Which sentence is used for apologising to someone?

I guess it would depend on how many things you are apologizing for.
I would use the first in most situations, if it's just one apology, the first is fine. But I suppose if you had a list of sins.. the second one would work.

In addition to what Ceili said, I would add that the plural form is used even when it's a one offence deal. Maybe using the plural form is equivalent to saying sorry a number of times.

Hi JTT

Where I live, even if it is saying sorry for the first time, it is "Accept my apologies." So we have been making the mistake without realising it.

I didn't think that I suggested that using the plural was a mistake, Ms Tan. If I gave you that impression, I'm sorry for it most certainly isn't.

Either could be used for a one offence or even if the person believed that there were multiple offences.

Thanks, JTT, for the clarification.