I have been asked by a client to take some product photographs of some vintage mirrors that they would like to sell and I am struggling with getting good clear shots of the mirrors. As such, what is the best way to approach photographing them such the mirror does not appear to be washed out or contain any unwanted reflections?
As Rob said, a tilt-shift lens is ideal.
I talked to a product photographer who specifically mentioned it's use. Basically, you position the camera on a tripod just to the left or right of the mirror so it's out of the reflection. With a normal lens it will be obvious that it's taken at an angle but by using the shift function of the lens you're able to correct the perspective distortion (just like an architectural photographer) so it looks as if the photo was taken head-on.
This article has an example shot: http://www.popphoto.com/how-to/2011/05/complete-guide-to-tiltshift-photography?page=0,3
Mirrors reflect, there's nothing you can do about that (except spray the mirror with something but clients are unlikely to go for that).
All you can do is position the mirror and camera so that what's reflected is as unobjectionable as possible. A good idea would probably be to position the mirror to reflect a plain wall or ceiling. If you get further away and use a long focal length lens, the area reflected by the mirror will be much smaller which will make it easier to ensure there's no clutter (this also makes it easier to ensure your camera isn't reflected, whilst still viewing the mirror nearly straight on).
Remember that you will probably need two lights, one to light the frame, and one to light the area reflected by the mirror.
If the job has the budget for it, you could rent a shift-tilt lens. That would allow you to at least get the camera itself out of the reflection.
Take a look at the catalog pictures of any major store like target.com or walmart.com. The seem to have covered the reflective area with paper ( gradient might have been added in post processing). After covering the reflective area, you can just shoot it as you would any other product.
You can also do some green screening by covering the reflective area with some bright colored/fluorescent paper and then put whatever picture you find aesthetically pleasing during post processing.
Several answers have recommended a tilt-shift lens. If you don't happen to have access to one, note that you can get a very similar effect by correcting the perspective digitally.
Panorama stitching programs like hugin are very handy for this, as long as the picture includes suitable guide lines (see e.g. this tutorial). If not, you could always add some artificial ones around the edges of the scene and crop them out of the finished picture.
Read Light: Science and Magic, specifically the sections on "family of angles". Basically, you need to place the lights at an angle so they don't reflect into the camera. The book gives various techniques for managing reflections.
A tilt shift lens is nice but, well, photographing a mirror warrants also documenting how well it actually reflects things. You could consider either photographing it as part of a scene (namely accept the perspective distortion) or allow the camera to appear in the reflection as an example image. Of course, either implies using a narrow aperture since you need to have both the mirror frame as well as the mirror image appear reasonably sharp to make a good impression.
The alternative moving the cam out of the picture (and of course not using a tilt-shift lens) The tiled floor helps with recognizing the perspective but I still like this worse. That is the reason I did not bother wiping the mirror again after the cat walked over it. If this had been a serious contender, one would have had to reapply the window cleaner again, of course, and retake the picture.
I shoot mirrors all the time. I use two stands on either side of me "camera" hang black fabric on the stands. Shoot the pic and you will have a black image in the mirror. All you have to do is clean up a spot in center, or add a gradient over the black.
Retouch the subject whenever possible to reduce set-up, studio, and retouching in post-production.
Removable dulling spray is as innocuous as glass cleaner. Apply as and where needed and remove before returning to the subject's owner/supplier a CLEAN, soft, absorbent cloth duster.
The viewer "understands" the light matte "glow" finish as "reflective" and the spray does not detract from the form of metallic finish filigree or scrollwork.
Where you want the sharp lines that indicate a mirror-finish, carefully mask or avoid spraying those areas—The mirror frame, for example.