Changing poopy diapers, sleepless nights, daycare, teaching li'l Daredevil to read, etc. - not especially interesting.
Sadly, I think this COULD be done interestingly, especially with street-level heroes like Spidey or Luke Cage, who have to worry about their day-to-day finances, who is going to take care of the baby when both parents are working (and possibly also both superheroes), and it can range from dramatic to comedic to touching. It's just gotta be done right...
I agree it COULD be done well, but given the nature of superhero storytelling I doubt it ever will be, at least not with a mainstream hero. It might be done with some minor hero, or as a tale in and of itself from an indy publisher.
The adventures of the Plaid Avenger! Crimefighter by day, dad by night. Watch him don his Paternity Suit! See him open jars of baby food! Etc. etc.