Can a laser cut through a mirror?

Can a laser cut through a mirror if the laser is shot right on top of it, or will it just reflect?

Depends on the wavelength of the laser and the reflectivity of the mirror at that wavelength. No mirror is a perfect reflector, some amount of incident light is absorbed and converted to heat. If that small percentage carries enough energy to cut the mirror material, then cut it will.

QED nailed it. Also observe that nothing is a perfect mirror at all wavelengths, and that some mirrors will actually absorb a lot of light from lasers at wavelengths where the lighht isn’t reflected.

If the laser power is high enough, the problem becomes find a mirror that will last, since even specks of dirt on the surface will absorb enough to cause the mirror coating to deteriorate, which will make it absorb more, which will make it deteriorate, and so on. I have a lot of spoiled high power laser mirrors.
That said, it’s not the most efficient way to cut your mirror. I never understood what Goldfinger was doing trying to cut that slab of gold with what was obviously a hopped-up Ruby laser. He would’ve been better off with a saw. But that wouldn’t look as cool.

Front surface mirror or back surface mirror? The question about the front surface mirror was answered. For a back surface mirror, like those in your bathroom, the beam goes through the glass twice. What to you think?

I guess the question’s been answered, but it’s surprising how much heat a mirror can absorb. An aluminum coating reflects about 92%, which means almost 1/10 of the energy is absorbed by the mirror. There are high-reflecitivity coatings that can reflect 99% or better, but those tend to be very fragile. The 1% of absorbed light will quickly degrade the coating, and as it degrades, its reflectivity will drop.

This is a real problem for solar telescopes, specifically the heat-rejection mirror (donut-shaped mirror right at the focus, which passes through light from the region you’re observing and reflects the rest). Glass mirrors don’t stand a chance; any type of coating will degrade from heat and melt away. Solid metal mirrors are better; metal has much better thermal conductivity so it’s possible to keep the mirror surface cool. I believe some larger solar telescopes use water-cooled metal mirrors.