Post-Crisis Superman and Lois Lane starting from 1996. Initially, their marriage was undone when Flashpoint rebooted the DC Universe and erased them from continuity. Convergence brought these versions of Lois and Clark back by revealing they were trapped in one of the many cities Brainiac saved from erasure. When Convergence ended, the Kents and their new son are now living in the current DCU and have been for nine years, as seen in Superman: Lois and Clark and Superman (Rebirth).
Both sets of Superman's parents. His birth parents - Jor-El and Lara - were very close. In some continuities, Lara refused to leave Krypton because she would have to leave her husband behind. His foster parents Jonathan and Martha Kent are a very loving, very close-knit couple who have been married for decades.
Kara's birth parents - Zor-El and Alura - love each other dearly. In the Post-Crisis universe, Alura was completely shattered when her husband died.
Her foster parents (Fred and Edna Danvers in the Pre-Crisis universe; Jeremiah and Eliza Danvers in the Post-Flashpoint 'verse) are also very close-knit. Jeremiah and Eliza have been married for years and they think nothing of flirting to their boss's face while they are working.
In the Many Happy Returns storyline, New Earth Linda Danvers - a Supergirl who was human and unrelated to Clark Kent - spent five years married to Earth-One Superman after switching places with his cousin Kara.
ElfQuest has a lot of happily married couples, despite the elves' open sexuality. Among the elves there are Redlance and Nightfall, Strongbow and Moonshade, Woodlock and Rainsong, and Cutter and Leetah. You could also add Bearclaw and Joyleaf - they do have some pretty fierce quarrels, but always make up afterward. Also the trolls Picknose and Oddbit (really!) and the humans Nonna and Adar. In the event of romantic rivalries, forming threesomes is considered a perfectly acceptable, if relatively rare, solution.
What's more, in a sense Cutter and Skywise are "married" - yeah, they're both guys (though remember, in ElfQuestEveryone Is Bi) but in many respects they act like a married couple.
All of which is not surprising when you remember that ElfQuest was created by Wendy and Richard Pini, who've been happily married for over three decades.
There seems to be a tradition of this in the Flash legacy; see also Wally West and Linda Park, and Jay and Joan Garrick, who have possibly the longest-standing marriage in the DCU.
Retconned out in the New 52. Now Patty Spivot is the third point of a triangle, and the one Barry's actually dated.
The Hawkmen and their partners (both at work and at play, as it were) used to have a similar happy relationship. Pre-Crisis, Carter and Shiera Hall of Earth-2 were married for about the same length of time as the Garricks, and used to go off on archaeological expeditions together in between JSA meetings. Likewise on Earth-1, Katar and Shayera Hol were married right from their creation. Post-Crisis (and post-John Ostrander and Hawkworld), Hawk continuity is just a mess, but Carter and the new Hawkgirl still get together fairly often. In fact the very first Hawkman and Hawkgirl's corpses are the central power battery of the Star Sapphire Corps (who are fueled by the Power of Love)
Spider-Man: Peter Parker and Mary Jane until the One More Day comic. They're also married in the newspaper strips, and have now gone past the twenty-five year mark. This is parodied in Ultimate Spider-Man, in which Mary Jane and Peter are so close that Liz Allen starts to refer to Mary Jane as "Mary Jane Watson-Parker" because "You two are totally married!" Peter even tells MJ at one point that when they become old enough he would marry her.
MJ and Peter are still married in the Renew Your Vows 'verse, and have a daughter.
Fantastic Four: Reed Richards and Sue Storm-Richards tied the knot in Fantastic Four Annual #3, back in 1965, and they've been together ever since despite bumps. Namor (among others) has had an on-again-off-again Stalker with a Crush thing on Sue for a long time, and it hasn't always been completely one-sided.
Mystique and Destiny, Rogue's adoptive parents, had an amazingly long loving relationship that began when the two women met in the 1890s and only ended when Destiny gave her life so that Mystique could live. Although it is known that through some unexplained circumstances they were separated for unspecified times, there was never seen a cross word between them. Pretty good showing, especially considering they're both villains.
In The Authority, Apollo and Midnighter. Interesting in that they're both gay superheroes who are happily married. Frequently played with, too, as they still have squabbles, and the kid-equals-stability thing was utterly subverted in the Revolutions arc (though that was mostly down to Manipulative Bastard Arch-Villain Bendix). However, they've been together for a long, long time, and no matter what crap gets thrown at them, their relationship is consistently a very loving one.
Archie's parents, Betty's parents, Jughead's parents... even Veronica's parents, despite being mbillionaires, aren't using that money as an excuse to become a stereotypically disconnected rich couple.
Reggie may be a jerk, but he can't blame it on a dysfunctional family.
The only exception was Midge crying because her parents were on the verge of divorce and even then she said they would stay together "with counseling".
This is such a major point in the Animal Man comics that it affects his costuming. He adds a denim jacket over his regular skintight costume so he can carry around his keys and notes from his wife. He's the only superhero with "bring home milk" on his crimefighting agenda.
Teddy Altman (Hulkling) and Billy Kaplan (Wiccan) are well on their way to filling this trope. They're already the longest-lasting couple on the team, and they've been engaged since they were sixteen. Even a universe-ending parasite and mutual fears that Billy was accidentally manipulating Teddy with his reality warping powers ultimately weren't enough to break them apart, with their relationship stronger than ever after working things through.
Same goes for Karolina and Xavin over at Runawaysalthough as of recently Xavin was Put on a Bus. The six couples that make up the Pride also count - while they're all murdering, thieving supervillains, their marriages and family lives are quite healthy (except for the way the Steins treat Chase, but even then, they love him and are willing to die for him).
In Fables Beauty and Beast definitely qualify, despite the occasional bickerings (as Beauty points out, you can't expect a marriage to last for thousands of years without quarreling). Snow White and Bigby also qualify later.
Swedish comic book Bamse has the eponymous character happily married with four children.
The Legion of Super-Heroes: Lightning Lad and Saturn Girl, who were long-standing sweethearts since the Silver Age and the second couple to be married. When Saturn Girl became pregnant with twin boys, she was the one who decided to go back on active duty while Lightning Lad chose to stay at home to look after the boys as a House Husband. Their status as happily married was briefly undone during the tenure of Geoff Johns as writer for what was considered the original Legion, with Saturn Girl trying to keep her husband under control while he kept snapping at her for never taking his side. When Paul Levitz came back as Legion writer, Garth's anger was toned down and the Ranzzs have gone back to being a supportive couple, this time with both of them retiring from active duty to raise their boys on Garth's home planet Winath.
Also from the Legion are Bouncing Boy and Triplicate Girl, who were the first Legion couple to get married. In every continuity where they're married, they've never been shown as anything but absolutely crazy for each other.
Cartoonist Charles Addams invariably depicted marriage as a literal battlefield, with the notable exception of his reoccurring characters who became The Addams Family: in their own very unique way, proto-Gomez and Morticia are a loving and devoted couple.
In PS238, Malphast's parents are quite happy together, although, given they're personifications of Chaos and Order, it's not entirely clear if they're actually married. But if they were humans, they'd probably be a model family.
Appears fairly frequently in Astro City, due to its overall optimistic nature.
From the First Family, there's Natalie Furst to Rex, and Nick Furst to Darcy. They've all had super-heroic kids of their own.
Zachary Johnson, the second Jack-In-The-Box, is married to Tamra Dixon, a local television news anchor.
Michael Hendrie (M.P.H.) is briefly mentioned as being happily married to a woman named Sally. Their relationship is not shown, but given how M.P.H. has been seen treating his girlfriends, it's hard to imagine them having any sort of drama.
Count Max and Ruby, the Vampire King and Queen of the Night from Scary Godmother, are head over for heels for each other. It's a case of Opposites Attract: Max prefers things old fashioned and likes adhering to traditional ways while Ruby is something of a modern woman who has no trouble adjusting to change. In one comic the two have a brief fight over traditional versus modern ways, and while separate, they see things about both lifestyles they don't find attractive at all. Max is followed by fan girls who want to obey his every command, which Max is reminded is something that was considered an old fashioned trait in women but which he thinks is disgusting. Ruby is accosted by several "modern" vampires who make rude comments and treat her disrespectfully. This also reminds Max and Ruby of why they love each other. Max thinks Ruby is vibrant and energetic and is attracted to her determination and ability to think for herself. Ruby loves Max's respectful demeanor and the way he both genuinely compliments her and treats her like a goddess.
Nathaniel Adam was happily married to Angela Adam until the experiment that either blasted him eighteen years into the future and left everyone thinking he had died, or until he died for eighteen years and then came back to life (what really happened is open to interpretation). She had remarried a man he hated and then died of cancer in the meantime, and his learning to reconcile with her memory is a major part of his character arc over the course of the series.
Circles: Paulie and Douglas, a gay example played absolutely straight. After Douglas broke up with his girlfriend, she never told him she was pregnant and had their son until after she died and gave up custody. Douglas' son had no hard feelings and still liked his dad especially because he saved his mom from a loveless marriage.