"Sorry, but I expect my kids to be more helpful." — My mother about three hours after yelling at us for pretty much anything about anything, mildly cut and censored.
This is the kind of non-apology I deal with on a near-daily basis, where my mother decides to only start with a "sorry", but it's really just blaming us more for anything about anything, which is usually about someone not doing some particular thing around the house, even if they do everything else.
It's always the "but" that makes it offensive, and they try to pass it off as an apology.
Of course, not only do I see it a lot, but I make it very clear that I'm not actually apologizing for being offensive when I do it. From me, it's more like, "I didn't mean to actually insult you; I just don't know how to put what I [tried to have] said [constructively] nicely... so I didn't." I was raised in a "do unto others as you would like done to yourself" house, too, so I'm expecting that they should be just as straight with me for the same purpose.