Inducing a "cuspidal" repn from SL(2,o) produces a finite sum of supercuspidals of SL(2,F). The easiest "cuspidal" repns of SL(2,o) are the ones that factor through SL(2,k), where k is the residue field. The "cuspidal" repns of SL(2,k) can be quasi-explicitly produced via the finite-field version of the Weil/theta pairing, inducing non-trivial characters from a k-not-split $O(2)$ (corresponding to the unique quadratic extension of $k$). Even a simple counting procedure easily shows that induced repns cannot account for all the irreducibles of SL(2,k), so we know that "cuspidal" ones must be there. The Weil/theta correspondence trick happens to produce them.

This kind of discussion already appeared a long time ago, I think in Jacquet's 1970 Montecatini lectures. In more recent times, work of Kutzko et al classifies supercuspidals of GL(n).