When I was on Mat leave, if the kids had a bad night, i would do the bulk of the stuff and leave Dh to sleep, as at the end of the day he did need sleep more than I did. But I would probably do practically nothing housework wise the next day if I was super tired and Dh would get home, take over for the kids, allowing for me to go upstairs even for an hours quiet if not a nap, while he cooked dinner.
Once back at work, we would tend to take turns.
Now, if I am on an early shift, ie up at 6 am, and the kids are bad in the night, DH will deal with them and leave me to sleep (as he work from home most of the time and does not have to do set hours, so can have an hour nap or so midday if he needed)
If I am on a later shift again we both share. If Dh has to be in the office, and I am on a day off the next day, then I deal with it leaving him to sleep.
So in our house it changes, depending on who has to be up earliest or who has a day off etc. If I am on an early, I am up at 6, if he is in the office he has to be up at 5 for a fairly lengthy drive so we just leave each other to sleep if that is the case.