Can particles be in a superposition of times as well as positions? We often talk about the various possible positions a particle can have upon measurement according to the probability density. But owing to the profound link of space and time in relativity, why do you never hear of possible temporal superpositions?
 A: A particle can be in a superposition of different states. Superposition of positions is not really a thing in our models.
When measuring a particle's position, you get a spread of values even in a pure state  in the general case -- but that is not called superposition!
OK, so now you may ask, do we get a spread of values when measuring a particle's time? Well, to answer that we need to know what exactly do you mean by measuring a particle's time. We need the description of your experimental setup.
A: I don't think this question has an exciting answer.
Strictly speaking, superposition refers to a wavefunction occupying multiple states at a particular time, so the question you've asked is "why can't you have multiple times at the same time", to which the answer is just:
given: $n>1$
then: $n \neq 1$

So, what if we just switch axes, and define superposition of times to mean a wavefunction occupying states at different times at a particular place?
The wavefunction is already defined as $\psi(x,t)$, so... we're done. The information about multiple states at different times is already there, no extra steps required.
Moving beyond a wavefunction, there's also nothing particularly quantum mechanical about an object existing in multiple times in the same place (this just describes any elapsed time for which average velocity was zero). "Superposition of times" in this regard is just our everyday experience of how location works.
Nor is there anything mysterious about multiple quantum states of the same object at different times. A photon interacts with an atom, an electron is promoted to a higher energy state at $t_0$, then decays to a lower energy state at some time t. There is a certain probability of decaying to the lower energy state at every given time $t>t_0$, all in (more or less) the same place.
A: Short and intuitively (since I'm not a physicist):
Superposition at one point in time means that an object could be at multiple locations, at one given point in time (or in other words: we don't know where it is right now, or instead of tracking a fixed x/y/z location for the object, we keep track of a complex probability distribution).
Superposition at one point in space would mean that a particle could enter a specific location in space at multiple times.
The first case is interesting, mostly because it is a quantum effect which contradicts our real life, every day experiences, and because it is much more complicated in mathematical terms.
The second case is uninteresting because it is trivial. Things move around, and both in the quantum world as well as our macro world, can return to the same location they were in before.
A:  A permanent particle in the schrödinger picture 
First, consider a stable particle. The picture below shows two possible worldlines for the particle, one in blue and one in green:

Now, consider a quantum superposition of these two possibilities. In the schrödinger picture, the state is parameterized by time. In a relativistic model, a state is associated with a spacelike hypersurface. Two different spacelike hypersurfaces are shown in the figure, one represented by the solid black line, and another represented by the dashed black line. Both of these spacelike hypersurfaces necessarily intersect both of the worldlines somewhere, so in both cases we would describe the state as representing a particle in a superposition of two different positions. The two hypersurfaces have different notions of which events are simultaneous, just like in classical special relativity, but no matter which frame we consider, we always have a superposition of different positions.
 A temporary particle in the schrödinger picture 
Instead of a stable particle that always existed, consider a particle (say, a muon) that is produced through the decay of some other particle and then lives only a short time. I won't re-draw the picture because it should be obvious: by considering a superposition of two different positions for the parent particle, we can end up with a superposition like
$$
 |\text{the muon exists now}\rangle
 +
 |\text{the muon doesn't exist yet}\rangle.
$$
Even if the time-of-decay would be independent of the parent particle's location in one frame, it won't be independent of the parent particle's location in other frames.
 A temporary particle in the heisenberg picture 
In the schrödinger picture, we might not feel comfortable calling that a "superposition of time windows in which the muon exists." But now consider the heisenberg picture, where all time-dependence is carried by the observables instead of by the state. Then the state is not associated with any particular spacelike hypersurface. It describes something more like the whole history of the system. Different places/times are probed by considering different observables instead of by considering different states.$^\dagger$
$^\dagger$ Since we're considering relativistic quantum theory, we should use quantum field theory. In quantum field theory, observables are not tied to particles. Observables are tied to spacetime instead.
In this case, describing the state as a superposition of time-windows in which the muon exists makes more sense. It's not a superposition of different times, because time is just a parameter. It's not a superposition of different places, either, for the same reason. In quantum field theory in the heisenberg picture, time and space are both used only to parameterize the theory's observables. What we're describing is a superposition of two different physical situations, one in which the muon exists only in one region of spacetime and another in which it exists only in a different region of spacetime:
$$
 |\text{the muon exists only in $A$}\rangle
 +
 |\text{the muon exists only in $B$}\rangle,
$$
where $A$ and $B$ are two different regions of spacetime.
This illustrates a general rule: if we want to think of time and space more symmetrically, then we should use a formalism that treats them more symmetrically. Quantum field theory in the heisenberg picture does exactly that.
A: I'd like to split your question into 2:

*

*are there temporal superpositions?

*can we measure temporal superpositions?

Are there temporal superpositions?
This question has been answered by g-s already with: yes!
$\psi(t)$ can be understood as components of an
abstract vector $|\psi\rangle$ in a base of time eigenvectors $|t\rangle$ ("time representation").
$\delta(x-\xi)$ are the well-known amplitudes of position eigenvectors
in the position representation with position eigenvalues $\xi$. Similar you can think of
$\delta(t-\tau)$ being amplitudes of time eigenvectors
in the time representation with eigenvalues $\tau$. So a function depending on time in general is a superposition
of time eigenvectors with the amplitudes $\psi(\tau)$
$\psi(t) \sim \int_{-\infty}^{\infty} \mathrm d\tau \, \psi(\tau)\delta(t-\tau) $
Can we measure temporal superpositions?
To experience superpositions in space we must build an experiment that gives us such processes ("collapse of space superposition")
$\int \mathrm d\xi \, \psi(\xi)\delta(x-\xi) \quad \rightarrow \quad \delta(x-\xi_{measured})$
I.e. we observe "a particle" at position $\xi_{measured}$ with probability $|\psi(\xi)|^2$.
Can we build an experiment that gives us similar processes with t instead of x?
This is the difficult part of your question. According to the theory of relativity
we would expext "yes" as answer.
According to non-relativistic quantum mechanics we can't as already said by Níckolas Alves. There time behaves like time in classical mechanics.
Relativistic quantum field theory considers relativity (i.e. Poincare symmetry).
But when calculating transisition probabilities, time gets a special role again.
You might expect that probabilities are calculated by integration over space and time.
But in QFT you will get $\infty$ when doing so, as The_Sympathizer already pointed out.
Now my personal opinion: we are missing a theory that allows to calculate
transition probabilities between block universes. We are mixing physical times
with psychic times, this is our confusion. Physical times are indexes enumerating base vectors. Like
space positions they do not flow by themselves and they have no direction built-in.
Psychic time flows by itself and has a direction. I tried to further work out the idea in
About Psychic and Physical Time.
According to this idea the answer to the question
"Can we build an experiment that gives us similar processes with t instead of x?"
is just "you are this process". Your conscious perception delivers information about
"quantum numbers" like space positions, ... that are changing.
There is 1 quantum number that always changes in every of your perceptions. This 1 quantum
number you call "time". A Lorentz boost changes your perspective, so that your perception, i.e. your psychic time,
couples to a slightly different direction in Hilbert space, i.e. physical time.
