Where to draw the line between quantum mechanics theory and its interpretation(s)? I've completed a full year QM course (undergraduate level) and I am left confused on where to draw the line between quantum mechanics theory and its interpretation(s). I would personally like to stick to no interpretation at all, but since I do not know what is interpretation and what isn't, it is extremely hard to stick to this rule. Many introductory books do not mention if they use a particular interpretation at all, and I suspect they do use some interpretation(s) here and there, without any warning nor notice.
From what I have read on the Internet, the "collapse" or "reduction" of the wave function, is part of interpretations of QM. Not all interpretations assume there is even such a thing as a collapse of $\Psi$. Good, that's an easy one.
But what about what $\Psi$ represents for example? I've commonly read that its modulus squared represents the probability density of finding the particle(s) at a particular position(s) and time(s). But does such a description already assume an interpretation?
What about the QM postulates? Is there any interpretation hidden in one or more of these postulates?
I've read several Lubos Motl's posts (here on PSE and on his own blog) and to him (and apparently many others such as John Rennie and Zurek), $\Psi$ is entirely subjective and two observers of the same quantum system need not to use the same $\Psi$ to describe the system. But no mention of any interpretation is ever done. I suspect they use some interpretation to make such claims, but I couldn't get the information from skimming through many books (including one by Zurek called "Quantum theory and Measurement" which is a package of many QM papers and one such paper by London around page 250 seemed to agree with the Motl's description).
I have heard of the "Shut up and calculate!" approach, but I have read on Wikipedia that it's associated to the Copenhagen interpretation. Is that really so? 
I have read from the member alephzero that QM works perfectly well without any interpretation. Quoting him:

"Wave function collapse" is not part of QM. It is only part of some
  interpretations of QM (in particular, the Copenhagen interpretation).
  The fact that this interpretation is used in a lot of pop-science
  writing about QM doesn't make it an essential part of QM - to quote
  David Mermin, "just shut up and calculate!" Note: AFAIK there is no
  so-called "standard interpretation" of QM - it works perfectly well as
a theory of physics with no "interpretation" at all.

My question is, how on Earth do we draw the line between QM theory and its interpretations? The books seem completely blurry in that aspect and almost any other sources I could find too.
 A: Interpretation is whatever people don't have to agree on to have the same accurate predictions about the observable.
Classical mechanics is empirically wrong in ways quantum mechanics isn't. For example, only quantum mechanics predicts discrete energies for atomic electrons, and discrete changes in these energies from the absorption and emission of radiation. How do you get these energies? Empirically, you measure them; theoretically, you reduce it to a calculus problem. These agree; there's no "interpretation" at work there.
Meanwhile, there are experiments you can do that vary in their results from time to time, and the frequencies of the results are, again, available both empirically and theoretically. The latter comes from the same calculus apparatus. What's that? You have a formula for something called $\psi$, whose square modulus gets us the answers we want? Great, our theory is predictive (insofar as anything probabilistic deserves that label.)
But what's this $\psi$ that crops up in both of those exercises? Well, it's not a thing classical mechanics makes claims about, or experiments detect; so whatever answer you give to that question, it amounts to an interpretation of quantum mechanics. Oh, you need $\psi$ or some alternative to get the predictions, and the predictions are right; no-one disputes either of those statements. But when you ask what these items "are", or "what they do unobserved", that's interpretation.
Get 20 QM experts in the room, each of them subscribing to a different interpretation. They'll all make the same predictions about experiments' observable outcomes. And if, in an experiment that leaves an electron's position unmeasured, one of these experts says the electron is "somewhere specific we don't know", and another says the electron is "everywhere at once", and another that it "doesn't have a location", they've found something they disagree on. It's just not an observable thing.
This doesn't mean interpretation is bunkum, or interpretations are wrong, or you shouldn't think about interpretations. (Fun fact: philosophy of physics is not limited to awkward questions about quantum mechanics.) But since your question is about where the line exists between interpretations and the rest of a QM textbook's contents, well... see the bold sentence up top.
Trust me, I understand the urge to put as little philosophy into things as possible. I do, I love me some number-crunching. But that should cut both ways, i.e. you don't want too many philosophical opinions about how little philosophy physicists should be doing either. For example, "shut up and calculate" doesn't have to mean "don't have an interpretation"; to me it means, "it's 9 am and we're predicting experimental outcomes; you can wonder what's going on 'behind the scenes' when we're at the bar". (Or vice versa!)
"Philosophy" isn't necessarily worse than "physics". It's just you can discern which is which from the fact that we know better how to get everyone on the same page for some questions than for others. Maybe that's not a bad thing. You don't have to agree the lack of an interpretative consensus is "embarrassing", but it's worth knowing that consensus is lacking. 
A: There are various forms of the "Copenhagen interpretation," but most of them share the following feature: they are not really interpretations at all.  Typically, a crucial part of what an interpretation of quantum mechanics does is make ontological claims ("what is real"), which are notably absent in the formalism of quantum mechanics.  True, some interpretations (e.g. QBism) explicitly don't do this, but QBism still makes a strong statement about what the wavefunction is (a state of knowledge).  Copenhagen generally doesn't make such claims, so it is the "operational interpretation," if you will.
So the bottom line is: if you want to avoid interpretations, the Copenhagen "interpretation" is a good way to go.  The line you are thus choosing to not cross is essentially that of making ontological claims and ascribing meaning to the wavefunction, beyond as a tool to predict measurement outcomes.
A: IMO yours is an excellent question, well expressed. I dare say an honest question (for the first time I've upvoted a question). I wonder however if you'll receive satisfactory answers. I'll wait and see if your question will be put on hold as requiring opinions, not a mainstream physics answer.
I can't give an answer of mine, but only some notes.


*

*Don't believe who says you can use QM with no interpretation. This isn't true for any chapter of physics. In many cases interpretations are implied, assumed as obvious and unquestionable, but are always present.

*The rule "shut up and calculate" is void (IMHO). QM is a theory and you are requested to apply it to experiments. There is a metaphor (due to Hempel) I like very much: theory is a net suspended over reality's ocean. You need some wires attached to the net and fishing into the ocean.

*Try an exercise: consider the problems proposed to yourself and
look where and when the questions got out from mere mathematical
computing (find eigenvalues and eigenvectors, show whether some given operators commute, etc.). Carefully watch the wording of questions. There very likely an interpretation is present, explicitly or not.
A: This generated a lot of answers: Here I want to define "interpretation"
In quantum mechanics you have an interpretation if the mathematically computed predictions for experimental verification are the same with the universally taught at beginning courses Copenhagen interpretation.
Example: Bohm's  mechanics is an interpretation of non relativistic quantum mechanics, as it predicts the exact same numbers for experiments,
Another example is the many worlds interpretation,it uses   the mathematics of quantum field theory giving reality, branching off  to all possible probable outcomes of the calculations as existing worlds.
If/when an interpretation of quantum mechanics will come up with experimental predictions different than the Copenhagen  one, it stops being an interpretation and is up for validation or falsification. (it and the Copenhagen).
Here is the set of postulates common to all Copenhagen versions.
A: Interpretations are here to make sense of the formalism of QM. Whatever computations and symbolic manipulations you need to do to prepare and analyse an actual experiment is QM. Whatever conceptual images you use to develop an intuition about what is going on is an interpretation.
When people say "there is no collapse", this may be an interesting point they make at some level, but the fact remains that after an actual measurement you have to update the quantum state of your system. Whether one calls this "collapse" or something else does not matter, it is a required step that you just cannot do without in any practical way - so this is part of QM, not of any interpretation. 
Now the bare truth is that nobody knows what QM is really about; it is still largely a puzzle. People uncomfortable with this state of affairs push their favorite interpretation and pretend everything is settled, but it is not. Once it is settled, there will be no point in the whole idea of "interpretation". There is no "interpretation" of thermodynamics, there is no "interpretation" of Hamiltonian mechanics, because the conceptual frameworks of these theories do not hurt our brains the way QM currently does. The trouble with QM interpretations is that every single one of them is just completely unacceptable at the conceptual level for a whole class of people. 
For some detailed discussions about this, I recommend Laloe, 2004 and Pablo Echenique-Robba, 2013. See also Mermin, 2009 and Landsman, 2005.
A: Interpretations and theories can be many but the interpretations that result in experimental findings are what scientists strive for and then we have "valid" interpretations compared to the unproven interpretations and disproven interpretations.  I think your question is about the valid of which there are examples.  Schrodinger accurately predicted the energy levels of hydrogen using QM, QM is a probability based model with equally important constants and boundary conditions required to produce the results. The theory is not so great at predicting accurate energy levels for more complex atoms molecules but the basis of the wave function goes on to predict chemical behaviour (like Pauli's exclusion principle) and many other behaviours of atoms and molecules. 
QM goes on to try and explain the behaviour of light (photons) and this is a fascinating area where interpretations can get very confusing.  One of the big issues is that there was so much discussion and difference of opinion in the 1920s,1930s,1940s and this led to some very strong viewpoints.  And strong viewpoints lead to even more interpretations but also led to some silos being created.  Is it a wave .. is it a particle .... does it interfere .... collapse ... is it like water .... is it all just probability  ... classical or quantum.
Young's double slit experiment (YDSE) is an amazing example of great and poor interpretation and almost intentional confusion. Single photon experiments (1960s?) were  a great realization ... but not widely popularized, it challenged the knowledge of great historical scientists.  The notion of interference was challenged, 2 photons can not cancel yet we have a dark spot in the pattern? Feynman went on to do path integrals over all paths and eventually realized that only the shortest path that was n multiples of the wavelength was the acceptable one.  From there one can interpret that the dark spots are where no photon can fall and the bright ones where many fall.  Do we really need comments like "the photon interferes with itself"?  Or does the photon simply find its path in a probabilistic way bounded by its wave function (the n lambda multiples being an important constraint)?  
Finally take the beautiful quantum eraser experiment and the (delayed) Mach-Zender interferometer experiments and you will hear of many interpretations but only the "photon wave function" (PWF) provides a useful explanation. It seems that photons are always looking for viable paths and can change their mind in an instant ... i.e. seeing all possibilities but choosing the best one.  So does the PWF extend everywhere in space but disappears upon absorption (collapse) ...  I do not think anyone will really ever know.  But apparently the PWF is mathematically unlocalized ... which probably lends it to even more interpretations. 
IMO photons never interfere, they have valid paths or not.  I think many scientists would agree ...  but its not a popular notion or interpretation.
A: Quantum mechanics is clearly an effective theory which is by no means complete. It does not even comply with special relativity. As any effective theory you can think of it as a set of rules that allow you to make predictions. None of the predictions of QM within the validity range of this effective theory has been proven wrong, so we use it, but we know of course that it is incomplete. Since it is an effective theory only, IMHO it does not make a lot of sense to go for an interpretation. (You can of course develop some intuition, and as long as it is consistent with the rules of QM, this may help you to "guess" the predictions correctly prior to an exact computation, but I wouldn't call this an interpretation.) As long as you deal with QM as with any effective theory, namely to make predictions using it as a toolkit in its validity range, you will be fine. 
A: Bohmian Mechanics (a.k.a. pilot wave theory) may well predict the same results as the Copenhagen interpretation, but it uses different assumptions. In particular, it assumes a preferred reference frame so that it can have an absolute time ordering of events. Thus, Bohmian Mechanics denies the relativity of simultaneity required by special relativity. 
At what point do we stop calling something an alternate interpretation and start calling it an alternate theory?
