Found this eventually:
When a character gets dizzy, either stars or birds appear above his head.
Generally, the bird dizziness lasts a bit longer than the star dizziness...
It's a good way to judge whether you have enough time to run in and land a
combo. Of course I always try to regardless

On Super there are four kinds of dizzy animations:
These are grim-reapers, not skulls. As far as I know, they have nothing to
do with killing your opponent. The only difference between the symbols
floating above your head is how difficult it is to 'RECOVER' (to coin a CPU
phrase). Incidentially, here's the list in order of ascending difficulty.
Angels - Stars - Birdies - Grim Reapers.