I always liked the Marvel Swimsuit Spectaculars, though not for the reasons you think!
They were fun because they showed the Marvel Universe at play, in a "chilled out" moment. I always really liked "Day of the Life" stories that didn't necessarily feature supervillains but showed what the characters are like "after hours." It always made their lives seem real.
Plus there was some great art, by guys like Joe Sinnott. This particular image gets me misty-eyed because it shows what Thor would have looked like if it'd been inked by a polished pro like him and not the sloppy Vince Colletta. I've always suspected that part of the reason that the Thor Silver Age stories, despite their importance, have never been seen as "required reading" is because of how godawful the art looked under Vinnie.
DC never did anything similar to my knowledge, but I always liked that Paul Levitz and others, very skilled plotters, realized that you can't have Mordru or some other worldbeating menace show up all the time - it's frankly, exhausting, so after some major story arcs, they did stories that dealt with nontraditional challenges and character moments, where often a single punch isn't thrown or a proton beam isn't fired. The best example would be the issues after Great Darkness, which dealt with Mon-El and the horrible scars done to the planet Daxam, which was in essence laid waste by Darkseid. Another good example would be the wedding of Donna Troy.
One of the problems I have with the Grant Morrison run on JLA in the 1990s was that the pace of it was exhausting. Every single arc, everything in the universe as you know it is threatened. The source of the superthreat differed, but it was pretty much the same kind of story over and over - and even usually had the same kind of resolution, with either Batman or Superman saving the day at the 11th Hour.
Kurt Busiek's Astro City is one of those comics where, you read and it makes you think very differently about comics afterward. I think what Busiek was trying to do with that book is to show that what is so interesting about superhero comics isn't the battle and the asskicking or even the epic conflicts against villains, but the personal stories. So he wrote a comic that has almost an inverse relationship to the focus: battles are often covered in a single panel, whereas stories that are usually ignored in other books become the central theme.