Byrne-era, it wouldn't stain as his aura would eventually remove it (see the stains that came off way back when he was on Apokolips during Legends, for example).
Pre-CoIE, the guy 'washed' his uniform by flying through the sun...
Birthright, seems to shrug off stains.
NuAction - it would stain
NuCurrent - it won't stain
Lois and Clark - it stained (and the iron burned a hole in the cape... )