In terms of comics, DC was bodying Marvel's plot lines all through the '90s. No Man's Land was better than any Marvel story released at the time. They also had MUCH better shows in Batman TAS, Superman, and JL/JLU. Like mentioned earlier, Bats, Supes, and WW are still the most recognizable superheroes in the world.
If you want to talk their status on the big screen and in pop culture now, DC has fallen behind Marvel because of their weak leadership. Diane Nelson and Barry Meyer were averse to risk and wouldn't give the entire DC roster the shine it deserved and relied on Batman alone because the franchise sells. They refused to start a cinematic universe unless Nolan helmed it, as if the world began and ended with him.
Marvel succeeded because they were willing to take risks with characters who were previously B-grade like Iron Man, Thor, and the freaking Guardians of the Galaxy. DC gave one half-hearted shot with GL and ran. That's why Marvel is leading the charge for the genre.
It would've been a different story with Jim Lee at the helm.