Removing sink intact from cultured marble vanity-top

Look underneath and it should be pretty obvious,

sinks generally clamp to the counter.

I'm presuming that this is a sink that "drops in" to the counter top. If so, there should be clips underneath it holding it in place as Philo says.

You need to remove (or at least loosen) those clips, then remove any caulking around the perimeter of the sink, and then pry it up.

Use this tool, which you should be able to find in every hardware store, to pry the sink up:

It's actually called a "bee keeper's bar" but it's typically sold as a combination pry bar and scraper. It's good for this kind of work because it's got a thin enough edge to get under the lip of the sink.

Also, your sink may have silicone caulk around it. If so, look in the caulking aisle of any hardware store for a product called "Silicone-Be-Gone" to remove that old silicone completely, Silicone-Be-Gone doesn't dissolve the silicone caulk, it just makes it swell up and get soft so that it can be removed by mechanical means. If you're throwing the old counter top out, it doesn't matter if you scratch it with razor blades, but I would use something less aggressive to remove the silicone from your china sink, like the WHITE Scotchbrite pads sold in grocery stores for scouring delicate objects.

Normally, Scotchbrite pads have an abrasive impregnated right into the fiber they're made of, but the white pads don't have any abrasive at all in their nylon fiber.

Under mount sinks may just be glued up there with silicone sealant. Cut it out with a razor knife.