Mnev's universality corollaries, quantitative versions? Mnev's universality theorem claims that any semialgebraic set is the realization space of some oriented matroid. Moreover, the rank of the or matroid can be prescribed in advance.
1.-Are there interesting corollaries to Mnev's theorem?
I am aware of interesting algorithmic consequences.
Geometric consequences? 
Examples in which the theorem is used to prove that other moduli spaces can also be wild?
MacPherson's definition of "combinatorial differentiable manifolds" and oriented matroid bundles are based on a local system of oriented matroids over a simplicial complex.
Is there some implication from Mnev's theorem to the theory of combinatorial differentiable manifolds.
What about proofs that would be easy (or statemens that would be true) if realization spaces of oriented matroids where better behaved, say connected, or contractible.. 
2.-Are there quantitative versions of this theorem relating (say) the number and degrees of the defining polynomial (in)equalities or the betty numbers of the semialgebraic set with the rank and number of elements in the corresponding or-mat. 
 A: Here are some references http://www.pdmi.ras.ru/~mnev/bhu.html 
A: For your first question, you might be interested in Ravi Vakil's paper "Murphy's law in algebraic geometry". He uses Mnëv's theorem to show that a large family of moduli spaces which are known to have singularities are in fact "as singular as possible", by which he means that every possible type of singularity defined over $\mathrm{Spec}(\mathbb{Z})$ will appear at some point of the moduli space. 
Here's a different application. Kontsevich defined for every graph $G$ a hypersurface $Y_G$ in a way motivated by QFT and the theory of Feynmann integrals. Motivated by computer experiments, he suggested that period integrals on the $Y_G$ should always be multiple zeta values. I am not sure of the precise relationship here, but I believe that this is (at least morally) the same thing as stating that the cohomology of $Y_G$ contains only mixed Tate motives. This is a very strong condition to impose and would say that the cohomology of $Y_G$ is extremely special. In particular this would imply that the function $q \mapsto \#Y_G(\mathbf F_q)$ that counts the number of points on $Y_G$ over a finite field is always given by a polynomial in $q$. Belkale and Brosnan in "Matroids, motives and a conjecture of Kontsevich" disproved this conjecture in the strongest possible way: they showed that for ANY scheme $X$ of finite type over $\mathbf Z$, the function $q \mapsto \#Z(\mathbf F_q)$ is a finite linear combination of functions  $q \mapsto \#Y_G(\mathbf F_q)$ for graphs $G$. Their proof uses Mnëv's theorem in a crucial way. 
A: So the universality theorem by Mnev, shows that every semi-algebraic set is stably equivalent to the realization space of a oriented matroid or a chirotope. So stably equivalent is a slightly stronger statement than homotopy equivalent. And homotopy equivalent implies for instance that all betti numbers are the same. Which in turn implies (to give a simple example) that the number of connected components are the same.
(Thanks to Michael Dobbins, who explained this to me.)
best
Till
