Having trouble reducing the vibrations on my turntable setup

I agree with whoever said that banging on the plinth or the shelf as a test for isolation  is beside the point, at best.

John at Townshend Audio advised I employ four Seismic Pods in all applications. Being accustomed to using three cones (first the Mod Squad Tip Toe, then the far superior BDR carbon fiber and Golden Sound DH ceramic cones) under everything, I went instead with three.

I discovered that three was generally fine for electronics, and in my case with the Sound Anchor loudspeaker stands bolted onto my 5’ tall planars (two in front, one in the rear, with the rear having twice the weight capacity rating of the two in front. Logical, right?). But for my turntables, four was much more stable than three. With three, the fairly-high mass plinths and platters tended to lean one way or the other; with four, as stable as can be. That might not be the case with low-mass tables.

The Pods aren’t cheap, but imo the only designs providing greater isolation are the much more expensive active platforms intended for use under microscopes, mentioned above by lewm. I've seen the Herzan in action, and if I had the dough I’d buy a dozen! The ingenious design element in the Pod (courtesy of Max Townshend) is the generally-misunderstood damping provided by the rubber sleeve and air release valve that are built around the inner spring. Watch the YouTube videos in which Max explains and demonstrates the design and effectiveness of his creation. Then all will be clear.

The rationale for using only three footers under any single piece of gear is that three points define a plane. Whereas when you add a fourth point you don’t necessarily have a plane. Hence the device can rock back-and-forth on any two of its 4 points. Or, one of 4 is not bearing its share of the mass which shifts a greater than average burden to another bearing. I agree that mounting a 100 pound turntable on three points is a bit scary, but it does give confidence that the turntable is nesting on a plane. After that it’s relatively easy to level the device by adjusting the height of one footer. 

I highly recommend (via my own experience) the Symposium Acoustics products.  They make a platform specific for turntables and to solve the very problem that you describe.  It's called the Segue ISO.  There are several standard sizes to choose from and also they can be made to order in a specific size of your choosing.  There are springs on the underside of the platform and you can choose to have the correct springs for the weight of your turntable.  The platform performs at it's best when combined with Symposium Acoustics Rollerblock products.  Those would be in place of your spiked feet.  Contact the company and speak with Peter (company president/owner).  He is extremely knowledgeable and helpful and will steer you properly.  The products are fairly priced and deliver an outstanding performance value.
Symposium Acoustics: (symposiumusa.com)
Best to you on your quest.

I would make sure that your tonearm and table are properly grounded and that you put it on an isolated wood stand with soft feet .