This is a problem I actually considered a couple years ago. It stemmed from a discussion I had with my suite-mates my second year at Argonne.

Under what condition will a "table" be stable?

The question came up as the dining table provided to us in our room happened to be unstable: it rocks back and forth and sits on three legs at a time. My proposition was that the only way to design a table that will not wobble is to have one with only three legs. And conversely, any table with three non-colinear legs must be stable. My reasoning is simple: three points define a plane. Given a plane (the floor), and three vectors oriented in the upward direction, such that the base-points of the vectors are not co-linear and the end-points of the vectors are not co-linear, such a configuration uniquely identifies a plane. And hence can not wobble.

Well, DJP was easily convinced. By our suite-mate Jim, who just loved to squabble with DJP and I, was not. And he (with help from amused suite-mates) tried to come up with counter-examples that showed me wrong. They came up with all kinds of seesaw like structures and pathological counter-examples that, in the end, I could only phrase my statement generically:

Consider the configuration space of tables with N legs in R^d. (The definition of tables and legs omitted here.) If N=d, then under the standard Lebesgue measure, the set of unstable tables are of zero measure. If N≠d, then under the Lebesgue measure, the set of stable tables are of zero measure.

or, I could use an idealized version of tables

Define the table-top T to be a compact, connected subset of the hyperplane x_d=0 in R^d. Define a leg to be a line segment (e,b] in the lower half plane with e in T. A table is defined as the ordered pair (T, L), where L is a finite collection of legs. A table "sits" on a hyperplane X of R^d if X doesn't intersect T, and if X intersects members of L only on the end-point b.

and state some more definite theorem to that effect. But that would take a while to reproduce, so I'll omit it here.

The point is, apparently I was not the only person troubled by the problem of wobbling tables. Andre Martin of CERN was also concerned. And so he did some calculations and produced a paper on the "Stability of four-feet tables". His solution is highly specific: only for four-legged square tables on ground of at most 15 degrees. Furthermore, his demonstration, rather than showing generic stability, was for the existence of a "rotation" of the table such that all four feet would simultaneously touch the ground.