Wall mount. Preferably on an outside wall because they tend to be stiffer than an interior partition wall. Make sure you lag bolt into the studs; if necessary, remove the drywall or plaster and put blocking inside the wall (that's wood that goes horizontally between the studs) and then re-drywall. DO NOT put up a wall shelf with toggle bolts through the drywall or plaster! If you can mount your shelf on a masonry wall, using lag bolts into anchors set in quik-crete, that's the best:~)
For a quick fix before you have time to do the wall shelf thing: Go under your house in the crawl space and determine where the floor joists are. Make sure you position the feet of your rack right over, or close to, each of two joists. (This is assuming the joists run perpendicular to the wall behind your rack.) DO NOT straddle one joist!
If the joists run parallel to the wall behind your rack, the same advise applies but you may have to pull the rack out from the wall to make it work. In both cases, you can "spot-stiffen" your floor by getting some concrete piers at Home Despot (they look like truncated pyramids about 10" high) place them on the ground in the crawl space right under the joists where the rack is located, and using short lengths of 2x4 cut 1/2" longer than the distance from the top of the pier to the bottom of the joist(s) hammer them into place vertically like little columns under each joist where the rack is. This will stiffen the floor at that location and (hopefully) keep your tonearm from bouncing -- which BTW is definitely not good for the cartridge suspension.