I'm looking to replace our current drop in sink with another drop in. I noticed that the current sink doesn't appear to be secured in any way other than I assume there is some caulk to connect with the granite. Is this normal? I was expecting there to be something underneath the sink securing it.

There is probably only support underneath if it is an under mount sink. Otherwise the lip above the counter is the only think keeping it there, and the silicone around the edge is preventing liquid from from going down below.

Mine is held on with some clips/clamps that grab the underside of the granite countertop to hold it down. They were a huge pain in the butt to get to, especially on the back of the sink and the side that's closest to the dishwasher. I bet plenty of plumbers don't mess with actually using them. By the time you hang the disposal and caulk it I doubt the sink is going to float off.