What kind of counters are you wanting? Undermount works with stone counter..There really isn't any disadvantages to either..

I go with an undermount sink whenever possible. You don't have that sink edge which can catch dirt and grime and can get pretty gross over time. With an undermount you can wipe crumbs, etc directly into the sink. Its a cleaner and more elegant look. However, its a little bit more involved to install. The installer has to screw the sink to the underside of the counter and the visible sink opening/counter edge has to be cut perfectly. A drop in sink is an easier install and the cutout doesn't need perfection. If you have the choice, do undermount! The look is definitely more upgrade.

Frances Temple-West explained it beautifully!

Undermount is a more high-end look, and it doesn't break up the flow of the counter quite as much as a drop-in sink. Splashes are also much more easily maintained and kept in an undermount sink. With a drop-in sink, splashes from hand washing dishes, etc. are much more likely to splash and run onto the counter because of the lip.