It is hard to find analogues of MA in which aleph_1 is replaced by the successor of a singular cardinal because
a) The consequences of MA-like axioms have large consistency strength
b) There is no satisfactory analogue of finite support ccc iteration

Dzamonja and Shelah found an ingenious approach to proving results of this general kind. I will outline their work and then describe some recent joint work with Dzamonja and Morgan, aimed at bringing results of this kind down to aleph_{omega+1}