Are there scattering targets other than nuclei, protons or electrons in experimental particle physics? I am not too familiar with particle physics, so maybe I missed something. Typical scattering targets seem to be nuclei, protons, electrons, i.e. stable targets, which of course makes some sense. Have there ever been scattering experiments involving two (moderately) unstable partners, e.g. muon - muon or muon-charged pion scattering?
 A: The problem with fixed-target experiments it that you have to make the target, then install the target, then irradiate the target with your beam.  If your target is short-lived, the timescale for each of these steps becomes more challenging.
For example, in positron-emission tomography, the positron source is usually fluorine-18, which has a two-hour half-life.  That means that a medical facility with a PET imager must also have a cyclotron on-site to produce the radioactive fluorine, perform the chemical preparation that incorporates the fluorine into a sugar-like molecule, and dose the patient with this sugar with the appropriate lead time before their imaging treatment.  If the fluorine-18 were produced across town from the imaging site, the facility might find themselves in the stupid position of being unable to reliably schedule PET scans during rush-hour traffic.
Fluorine works for PET imaging because a small amount of the radioisotope can be isolated chemically, dissolved in saline, and concentrated by the normal biological processes in the body.  But if you were to put that saline-buffered fluorinated sugar in the target chamber of an accelerator, the accelerator beam would interact entirely with the stable particles that make up the saline solution and the non-fluorine parts of the sugar molecule --- not because those particles are special, but just because they are numerous.  If you want a fixed target for an accelerator that's easy to analyze, you want that target to be made substantially of one material.  But transmuting a chemically-significant amount of mass from a stable element to an unstable element --- even from a stable element to a different stable element --- is a hard problem.  (Previously, previously, previously.)  No one is ever going to construct a nugget of francium, hammer that nugget into a foil, install that foil in a target chamber, and irradiate it to see what happens, because you couldn't do all of those things during francium's twenty-minute half-life --- even if you could make and isolate the stuff.
Generally, if you're interested in the interaction between an unstable particle and a stable particle, the solution is to generate the unstable particles in the beam.  The rare-isotope people, who are interested in superheavy elements and in isotopes with wacky proton-to-neutron ratios, do this by creating their short-lived particles and reaccelerating them.  For instance, if you send a beam of calcium ($Z=20$) ions onto a target that's a thin uranium ($Z=92$) foil, some of the interactions will be fusion events that send fast-moving, excited nuclei with $Z\approx 110$ out the downstream side of the foil.  Those can be collected in a secondary accelerator, sorted by charge-to-mass ratio by cleverly using steering magnets, and stopped in some calorimeter that collects all of the decay products, with a timescale of a few microseconds.
Re-acceleration is also how beams of unstable fundamental particles are produced.  For example, to make a muon beam, you send a proton beam into some solid or liquid target.  The strong interaction generates lots of pions, which decay to muons much more rapidly than the muons decay to electrons.  I've never worked on one of these machines, but I have the impression that the operating principles are mostly the same as the rare-isotope reaccelerators:  once you've made your beam, there are options to focus, steer, clean out decay products by charge-to-mass separation, etc.
One way to study an interaction between two unstable particles, like muon-muon scattering, would be to fill two storage rings with the unstable particles and look for interactions where the beams are allowed to intersect.  That's essentially that the Large Electron-Positron collider (now upgraded to the Large Hadron Collider) did.  However at LEP and LHC, the particles in the beams are intrinsically stable:  the way that machine functions is to fill the storage rings with charged particles and then allow the stored particles to interact for a very long time.  On each trip around the storage ring, a given particle has a vanishingly small chance of participating in a beam-beam interaction.  If one of the storage rings were filled with unstable particles like muons, the ring would empty itself rapidly as the muons decayed.  Also the ejection of the decay products from the storage ring would create a lot of secondary radioactivity.  The success of the muon $g-2$ storage-ring experiment suggests that this sort of thing is possible, but as @dmckee says in a comment, it's been "just a matter of money" for a long time now.
For interactions between neutral unstable particles, your only option is to generate a chemically significant quantity of the particles at an energy you think is interesting.   There is an apocryphal story about a neutron-neutron scattering experiment where the neutron "target" was a second nuclear bomb, to be detonated at the same time as the neutron "source."  The story goes that the experiment was reviewed, approved, and conducted at an underground test site; the detonations were successfully synchronized and the data were collected, but a blast door failed and the data acquisition system was destroyed.  However, I've never found a description of this experiment in the unclassified literature, which means one of (a) I haven't looked hard enough, (b) it's a great story but it never happened, or (c) I wasn't supposed to have been told about it.
The short version of this answer is that, if you're interested in scattering between unstable species, you're probably looking at beam-beam interactions.  The long version of this answer is what happens when you start to think about making unstable targets; it's just a scratch on the surface of what's possible, but basically all of the ideas are fatally flawed.
A: There are other forms of baryonic matter besides the ones you've listed, e.g., white dwarfs and neutron stars. There are cases where particle physicists have gotten useful bounds on certain observables from this. For example, Giddings and Mangano rule out certain scenarios involving large extra dimensions because microscopic black holes would have destroyed all our white dwarfs and neutron stars.
There is also two-photon physics: https://en.wikipedia.org/wiki/Two-photon_physics
Giddings and Mangano 2008, "Astrophysical implications of hypothetical stable TeV-scale black holes,"https://arxiv.org/abs/0806.3381
