should I suspend a floor-standing turntable?

Wood blocks can be found at lumber stores cheap and granite slabs can be found in monument shops -- work out the best price there. You don't need one size of monument for sure :-)

After you select one of the above options, if there is still any uncertainty whether you have overcome all manner of vibrations, there is one more inexpensive step that I have found to complete the job.
Try positioning three or four Cryo'ed hardened steel compression springs named "Cryo Baby Promethean Mini Isolators" (from Machina Dynamica) between the turntable bottom and the support surface of wood or granite. And/or try a set beneath the wood or granite, itself.
You are now better protected, and may proceed to engage in safe sound.
Listening to some favorite recordings will tell you whether each step you take toward better isolation is actually delivering more natural and satisfying audio in your system.

..just remembered....another excellent way to support the turntable is to attach the stand to the wall using turnbuckles (at Home Depot). Tighten the rig and you'll be amazed at how much better things sound.

Stringreen, this certainly has not been my experience and neither is hanging from the ceiling. In one home I had earth below where the turntable was and used a three inch pipe with a screwable plate that really did help and in another house built on a slab putting the turntable on the floor was quite good. Elsewhere sheer mass was the answer. I once heard a system in Chicago where the guy got curbstones under his turntable. Each weighed about three hundred pounds.

Wall shelf worked great for me in Germany.
Germans don't use drywall .