I have installed quite a few mirrors with backlighting, but they were for the most part all mirrors that came with the back lighting which shines through on the frosted transparent portion.
As to what they use, some have been linear fluorescent, while others have been bi-pin halogen.
For the backlighting of the entire vanity, LED's (ribbon light, even inexpensive rope led light) or incandescent rope, depending on what you are going after, give a nice effect. Another option will be something like Seagull Ambiance LX track with xenon heads (usually used for undercab or cove lighting). There usually isn't enough room to install something like a linear fluorescent when they frame it out this way.
So are you back lighting the actual mirror where it is frosted, or are you backlighting behind the mirror/vanity so that the sides are illuminated?