Which one do you use most often/in which contexts?

I would be more inclined to use 'stariĝi' if there was a lack of volition on the part of the stander. Some examples from Reta Vortaro are "stariĝis ŝiaj haroj" and "teknikaj malfacilaĵoj stariĝis".

For a person standing up I'd almost always use 'ekstari', but it's not wrong to use the -iĝ form for that and you'll find plenty examples of its use in the literature.