What does a genuine "I'm sorry" sound like?

There are a zillion ways to say "I'm sorry" but there are few ways to do it well. I've heard plenty of bad ways. There was the time my sister-in-law apologized for refusing to allow my baby to sleep in their (as-yet-unborn) child's crib by telling me that, yes, she was sorry but that I didn't pay as much attention to their (now born) child as I should so, really, it was a draw.
There was the "I'm sorry I slept with your boyfriend and betrayed our friendship but you were a horrible girlfriend to him and therefore sort of deserve this" apology from one of my closest friends that came in the form of a seven-page letter outlining the ways in which she was sorry but, really, not.
There was my husband's "I'm sorry but you're really making a big deal out of this" apology when his mother, at our wedding, began rearranging the seating for my guests in order to move her friends closer to the action. (And, incidentally, I never did get an apology from her.)
So yeah...lots of lousy apologies. Plenty of non apologies.
There was, however, also my mother's apology, after getting sober, which has pretty much set the bar for apologies in my life because it was heartfelt and did what an apology should do, according to Lux Alptraum, author of Faking It: The Lies Women Tell About Sex—and the Truths They Reveal.
Alptraum suggests that an apology has three parts (she's referring in this case to the apology owed to us by tech companies that have violated public trust but hey, an apology is an apology, whether from Facebook or our mother-in-law). According to her, “A good apology says, number one, ‘This was bad, I recognize this was bad, and you are perfectly within your right to be hurt and angry and upset.’ Number two, a good apology says not just that harm was caused but that the harm was someone’s responsibility. And, ideally, number three, it shows growth and commitment to repair.”
My mother's apology (which she made more than once) generally went like this: "I am so sorry for the pain I caused you. If I could take back those days, I would. You should never have been put in that position. I'm doing everything I can to get better."
Accompanying her apology was also action. I saw how many AA meetings she was attending, I was privy to the reading she was doing to learn about addiction and sobriety. I knew that, as part of the 12-step philosophy, she wrote letters to others to "make amends".  
An apology can never undo the pain caused. And none of us are obliged to accept an apology. Sometimes an "I'm sorry" barely scratches the surface of the pain that's been caused. Consider a rapist or a murderer, a drunk driver or...an unfaithful husband.
But apologies can be a crucial first step.
And, done well, they can be a balm to our broken hearts.