  The global geometry of the moduli spaces of higher spin curves and their
birational classification is largely unknown for g >= 2 and r > 2. Using quite
related geometric constructions, we almost complete the picture of the known
results in genus g <= 4 showing the rationality of the moduli spaces of even
and odd 4-spin curves of genus 3, of odd spin curves of genus 4 and of 3-spin
curves of genus 4.
