We investigate the tangent semicone C and the Nash space N (the fiber of the Nash blowup) of an algebraic surface V (with singular locus S) in R³. We prove a structure theorem for N: there are finitely many "exceptional rays" in C so that N is the union of N(C) and the set of elements in N containing one of the exceptional rays. The set of elements in N containing an exceptional ray is semialgebraic, but can be disconnected and have discrete elements if the ray is tangent to S. Any ray not tangent to S, but along which C is singular, must be exceptional (except in one case), and the set of elements in N containing the exceptional ray is closed, connected and 1-dimensional, and we can give a lower bound on the size of this set.