It's a rental. With that in mind, my go to choice would be an undermount. Second would be a drop-in. Never ever would I put a vessel in a rental. Do consider that if the vanity was designed for a vessel that any intrusion below the countertop might interfere with vanity storage. Draw functionality, etc. One sink versus two? If the bathroom is small, that might lead you to one sink. One person washing at a time, more counter space, etc. Plus, with your budget being tight? One sink will save you. If you think you'll have trouble renting this place and you think you might need an edge, or if you're renting to a couple/family, then two sinks could give you an edge in a tight rental market. Sometimes it's all about the bling. Just make sure that before you go the two sink route that the vanity itself can take two undermount sinks, or two drop-in sinks, PLUS the plumbing, without completely losing the functionality of the top bank of drawers in the vanity. If an undermount will cause you to lose storage, then I'd minimize the amount of lost storage by going with one sink.

Apologies if you have already considered this, but with a shelf vanity, if the sink does not sit on top of the shelf, then the sink will start at the counter, and go below the shelf. Its a design consideration (your right hand photo shows it ) -- what do you want visible -- plumbing wise. The depth of the vanity shelf front may need to be adjusted so that the bottom of the sink and the plumbing does not show, or plumbing is selected that is attractive. The cottage-y example has a deeper front on it to hide the sink bottoms. Benjesbride has an excellent point about keeping stuff on shelves clean -- it will take a lot of cleaning to not continually collect dust (it'll show on the shelving, and be embedded in the towels...dust, splashes, etc.). Good luck!

I had the exact same problem. I picked out the vanity I wanted but did not think about the fact that it wasn't made to sit flush against the wall. I spent a month looking for something else, but in the end we decided to go with something shorter and have it not be against the wall. But we were going from 72" to 60" which is probably not workable in your mom's situation. I did find a lot of vanities that did have flat sides and could work in a corner, if you are willing to have the countertop fabricated. They just need to not have any trim detail on the top or bottom like the one you picked out does. I would just look at the ones without the pre-made tops. Because the pre-made top will still leave you with a slight gap because of the countertop overhang. This is an example: Avanity Modero which is sold everywhere, not just at Lowe's. The cabinet we ended up buying is an Avanity Madison which won't work for you, but I assume it is of similar quality. We are pretty happy with it. The drawers and doors have soft close but I do feel like the glides in my Ikea kitchen drawers are better quality. It is very pretty and I wanted a stained wood so I am really happy with our choice. If I had been willing to go with a painted finish, I think we might have stuck with the 72" Modero.

Not a big deal. Pull the new vanity forward (it looks like you have about 8" of room to the casing) and run the plumbing behind it. Bonus, it gives you an extra deep counter. You could then put a shelf 6 or 8 inches up, running the full length of the vanity if you like.