"I'm sorry for" vs. "I'm sorry about"

Solution 1:

Given just those two option (there are others, such as 'I'm sorry that...) it depends on what you want to apologise for.

To take two extremes, if it is something very specific, like arriving late for an appointment, you might say 'I'm sorry for being late', or 'Sorry I'm late'. It would be unusual to say 'Sorry about being late'

If it is that you want to apologise more generally for some arrangements that have gone completely haywire, you might say 'I'm sorry about the shambles on Thursday'. You could also possibly say 'I'm sorry for the shambles on Thursday'. Using 'for' in this case does suggest very slightly your taking more responsibility for what happened than if you said 'Sorry about the shambles'. The latter could just mean that you are sorry it happened, but don't take too much responsibility.

So to sum up, it largely depends on two things, how specific the apology is (with 'for' being for the very specific), and how much responsibility you take (with 'for' implying greater personal responsibility).