Do modern bathroom sinks (china) not use any mechanical fastenings to
hold them to the countertop? I have one that wiggles in the cutout
and there is nothing underneath to tighten. There is a bead of caulk
under the lip of the sink, but it isn't stuck to the countertop.

Any suggestions, or am I going to have to unhook the pipes and
"reglue" it?

Thanks,
Red