I think that apologies that are followed by a specific indication of what is going to be done differently can be very effective.
Instead of, "I'm sorry, you're right, I shouldn't have done that."
"I'm sorry that I ate all of the peanut butter, from now on I am going to make sure I save some or replace it so there is some left for the kids lunches."
I dunno...IMO the only way to exhibit that an apology is sincere is to a) acknowledge what was done wrong and b) do it differently the next time.
H used to apologize and couldn't even really tell me what he had apologized for...