  We continue previous works by various authors and study the birational
geometry of moduli spaces of stable rank-two vector bundles on surfaces with
Kodaira dimension $-\infty$. To this end, we express vector bundles as natural
extensions, by using two numerical invariants associated to vector bundles,
similar to the invariants defined by Brinzanescu and Stoia in the case of
minimal surfaces. We compute explicitly these natural extensions on blowups of
general points on a minimal surface. In the case of rational surfaces, we prove
that any irreducible component of a moduli space is either rational or stably
rational.
