Few people in the Marvel Universe can claim to be worthy of Mjolnir. Even fewer in the DC Universe can make that claim. But there are some who can, or even have, lifted Thor's hammer in DC.
Worthy
Superman
Back in the day when DC and Marvel actually used to partner on comics from time to time, a situation cropped up when dealing with Krona. It gave Superman the opportunity to wield both Thor's hammer and Captain America's shield at the same time. It's important to note that the rules around worthiness had been temporarily suspended to allow this to happen.
As soon as the fight was over, Superman couldn't lift the hammer anymore. But under normal circumstances without any weird interference, Superman is worthy of Mjolnir. He's not the biggest boy scout in DC for nothing.
Billy Batson
Captain Marvel, aka Shazam, is one of the most powerful heroes DC has to offer. He's on Superman's level, though Shazam's powers are derived from magic. But being powerful is not what makes Shazam worthy of Mjolnir. In fact, Shazam is not the one who is worthy of the hammer. Billy Batson is.
When the young man speaks the magic phrase SHAZAM, he gains the formidable powers of six immortal elders. Batson had to be worthy of this power to wield it. The same qualities that make Batson worthy of Shazam's power make him worthy of Thor's hammer.
Adam Strange
One of the key qualities Mjolnir seems to respond to is sacrifice and Adam Strange is the poster boy for it. An archaeologist working in Peru, Adam suddenly found himself transported to the planet Rann via Zeta-Beam. Once there, Adam started building a life as Rann's greatest hero and defender. He also found love with a young woman named Alanna.
The unfortunate reality is that the effects of the Zeta-Beam are temporary, returning Adam to the same point he left from when it wears off. He keeps taking Zeta-Beams back to Rann but is repeatedly torn from the planet and woman he loves. That is sacrifice.
John Stewart
As a Green Lantern, John Stewart struggled with being second to Hal Jordan. Though the two men built a strong friendship over the years, Jordan casts a wide shadow in the Green Lantern Corps. What sets Stewart apart from other Lanterns is how he handles adversity.
At one point when his ring was depowered, Stewart found himself jailed and tortured in another country. He overcame that, in part due to getting his powers back, and moved on from there. While not every decision he makes is the right one, Stewart has proven to be a hero of strong will and character, exactly what someone needs to hold Mjolnir.
Wonder Woman
During the Amalgam storyline when Marvel and DC faced off for the survival of their respective universes, Wonder Woman found herself in possession of Mjolnir. Not only did she hold it, but she was also able to access its full power and had the opportunity to use it in the battle against Storm. She chose not to, then lost the fight, which is the exact reason why she was worthy. Wonder Woman is an honorable, principled warrior, the type of warrior who would give up an unfair advantage even if it meant losing a fight. Few heroes would be willing to do that.
Not Worthy
Batman
The Dark Knight might be one of the least worthy heroes in any universe. While Batman does have some of the characteristics you think would allow him to lift Mjolnir, he also has committed a number of acts that disqualify him. This is a man who once kept extensive files on his allies in case he needed to take them down, files that were stolen and used against said allies.
Batman's dubious, manipulative relationships with many of his fellow heroes are the least of the issues that would keep him from wielding the hammer. That being said, Batman is also the least likely to even want the damn thing so this probably won't bother him.
Green Arrow
Brash. Arrogant. Egotistical. Not only is Green Arrow not worthy of lifting Mjolnir, but he might also be the living embodiment of the qualities that caused Thor to lose the hammer in the first place. No one is suggesting that Green Arrow isn't a great hero.
His service to the world and the universe as a whole as a member of the Justice League is not in question at all. But that doesn't change the reality of the man behind the mask. Oliver Queen is not the type of person Mjolnir is looking for, even though he is a hero. The man is often a jerk in the worst sense of the word.
Aquaman
As King of the Seven Seas, Aquaman has had to make some incredibly dubious choices to protect his kingdom. Those choices have caused problems with the Justice League as well as various government bodies on the surface world. None of that specifically makes Aquaman unworthy of Mjolnir's might.
What does make him unworthy is who he has more becomes in more recent years. More frequently, he plays the role of a deposed king and a hero that has fallen from grace. Aquaman is neither the hero nor the man he used to be, and certainly not worthy of Mjolnir.
Huntress
As part of the extended Bat-family, Huntress rides on the periphery of what Batman finds acceptable in a crime fighter. Bertinelli is known for her brutality on the streets of Gotham. She's also known for making detrimental decisions to her career, like trusting the man who turned out to be Hush with upgrades to her crime-fighting gear.
Huntress is a tough hero but she also rides the line in some situations. She's constantly trying to learn and do better, but she's also a little too preoccupied with Batman's approval. It would be surprising if she was able to lift the hammer.
Hal Jordan
Considering that Hal Jordan might be the greatest Green Lantern to ever wear a ring, you'd think that would put him at the front of the line to be worthy. Not so much. His run as Parallax is a huge black mark on his record. Sacrificing the rest of the Green Lantern Corps to get the power to resurrect his home city was a horrifying plan.
Even worse, he tried to change things by destroying the universe and rebuilding it in a manner he approved of. No matter how much good he did before that time or after it, Jordan can't come back from the Parallax incident. He will never be worthy.