Does $s/p$-polarized light remain $ s/p$-polarized after reflection? Suppose you have a beam of $s$-polarized light that reflects off of some medium. Is the reflected beam also $s$-polarized? What about with $p$-polarization? I'm assuming those nice properties of media (whatever they are) that one assumes in typical undergrad E&M and optics courses.
 A: It depends on the medium.
For a medium with scalar refractive index, the situation is as described by Roger Wood. Pure s-/p-polarization will keep their polarization. This is in some sense by construction, since they are chosen as the "eigenpolarizations" of the reflection response. Since s-/p-polarization typically have different reflection coefficients (except in some limiting cases such as normal incidence and grazing incidence), a mixed polarization will change upon reflection. E.g. if the incident electric field is given by
$$\mathbf{E}_\mathrm{incident} = \alpha_s\mathbf{E}^\mathrm{(incident)}_s + \alpha_p\mathbf{E}^\mathrm{(incident)}_p \,,$$
where $\alpha_{s,p}$ are the relative magnitudes of the two polarization components and $\mathbf{E}_{s,p}$ their corresponding electric field vectors, then the reflected field will be
$$\mathbf{E}_\mathrm{reflected} = r_s\alpha_s\mathbf{E}^\mathrm{(reflected)}_s + r_p\alpha_p\mathbf{E}^\mathrm{(reflected)}_p \,,$$
where $r_{s,p}$ is the reflection coefficient for each polarization. We then see that if $\alpha_{s}$ or $\alpha_p$ is zero, the polarization is conserved. Note that this still involves a rotation of the electric field vector for the p-polarization, since the propagation direction changes upon reflection.
For general media, however, it is possible to have a tensorial refractive index, which rotates polarization. Such media feature effects such as birefringence and they would also rotate the reflected polarization direction.
A: The polarization of the beam will remain the same.
Polarization is a geometrical property of transverse waves that describes the oscillation of the electric field.
It can be more generally be described by a state vector such as $$|\psi\rangle = z_1|\uparrow\rangle +z_2|\rightarrow\rangle$$
Where $|\uparrow\rangle$ is your s-polarization and $|\rightarrow\rangle$ is your p-polarization.
Usually, when your beam passes through different materials, these states are symmetrically affected leading to no relative phase difference. It is only when you pass through polarizers, liquid crystals or any other material which propagates one axis differently that this changes.
A typical mirror is not one of these materials, and will not affect polarization.
A: The measurement of changes in polarization upon reflection is called ellipsometry. Waves polarized exactly in p (parallel to the surface) or s (perpendiular to the surface) will be unchanged in polarization though they may reflect (and refract) at different magnitudes.  Light polarized at some angle in between p and s will generally have both magnitude and polarization changed. 
Daylight reflecting of water and the use of polaroid glasses illustrate some of the effects.  
A mirror made of dielectric will show these effects but metal surfaces generally do not show the effect to any great extent.
