Since the 616 Universe is the main continuity of Marvel, there is bound to be plenty of powerful characters there, but since our list of Omega Class Mutants only covers those classified as such and their home dimension, it doesn't include their status or current location. So for the sake of those who want to know, I'm gonna list the relevant data here.
Omega Class Mutants currently in the 616
Hope Summers (616) - Born in Cooperstown, Alaska of the 616 under the family name Spalding, this time hopping Omega class has the ability to copy the powers of mutants in close proximity to her. After the insane events of AvX, she is now living a "normal" life at the Jean Grey School.
Jean Grey (616) - The original Omega Mutant, it's unclear to me whether her status as Omega is a result of her own impressive telepathic and telekinetic power, or being the first incarnation of the Phoenix with it's vast extent of pyrokinetic, life giving and other abilities. While having been deceased for some time, several times, she is currently counted as alive, but doesn't seem to be interested in getting involved in human affairs any more, instead focusing on gathering the last fragments of the phoenix scattered across the 616 Universe.
Franklin Richards (616) - The mutant child of Reed and Sue Richards, he is the most powerful earthbound character in the Marvel Universe. Possessing the power to manipulate the very fabric of space and matter with psychic talents aside, he is now Number 5 of the Future Foundation.
Robert Drake (616) - An ice controlling member of the X-Men and one of the original five alongside Jean Grey. Bobby "Iceman" Drake can control his body make up, reforming it entirely from ice as well as creating multiple duplicates of himself. His ability to control temperature and water make him a force to be reckoned with, and yet he's still the fairly clownish character and current accountant for the Jean Grey School of Higher Learning.
Gabriel Summers (616) - Younger brother of Scott "Cyclops" Summers and Alex "Havok" Summers, "Vulcan" as he prefers to be called was captured alongside his parents by merit of still being in his mother's womb. Artificially aged by the Shi'ar tyrant D'ken, Gabriel became a powerful and evil warrior, conquering for the Shi'ar Empire using his energy manipulating powers. Capable of variable energy blasts, regeneration, siphoning energy, Gabriel is an incredibly powerful fighter. He eventually met his end fighting Black Bolt when the pair were caught in the blast of the T-Bomb.
Rachel Anne Summers (811) - The child of Scott Summers and the Phoenix Possessed Jean Grey from an alternate future. Rachel is a powerful psychic abilities, including telepathy, telekinesis and to a degree, time travel. While her powers are effectively a weaker version of the Phoenix, she does not seem to possess its pyrokinesis. She is currently a teacher at the Jean Grey School.
David Charles Haller (616) - The son of Charles Xavier and Gabrielle Haller, David is better known to the world at large as "Legion". David possesses the ability to spontaneously mutate himself, giving himself a vast number of varying and powerful abilities. However, he suffers from Dissociative Personality Disorder, creating a new personality for every ability he possesses. Whether this is his way of coping with his vast abilities, or a result of his past trauma is unclear. He is currently trying to achieve his father's dream through "Proactive methods".
Nathaniel Grey (295) - The artificial offspring of Scott Summers and Jean Grey in the Age of Apocalypse. He was an unstable psychic with previously vast psychic ability, he is now left with a weak telekinetic power. He is currently living with the New Mutants in Mission Street San Francisco.
Joshua Foley (616) - A former member of an Anti-mutant group, Josh possesses an untold potential for Biokinesis, manipulating the cells of living things for healing. He can cure various diseases, possibly to the degree of curing every existing disease. On the other hand, he can flip this ability, rapidly developing cancer and other illnesses in targets. The only limitation being his need for physical contact to enact his power. He helped Wolverine after the Schism, but where he has been since is unknown.
Quintavius "Quentin" Quirinius Quire (616) - Otherwise known as Kid Omega, Quentin possesses powerful psychic ability, particularly in his telepathic power, but also limited telekinesis. He is a intelligent and rebellious youth who was close to becoming a Super Villain, but was recently brought into the Jean Grey School in an attempt to reform him. Most recently he is infiltrating the Hellfire Academy.
Absolon Mercator (616) - First appearing as one of the 198, Mr. M had the power to to manipulate molecules. Near Omnipotent, Absolon was a surprisingly pacifistic man. Despite apparently being killed in a plan orchestrated by General Lazer. Despite this, his casket was later be found empty, having left the message "'Some things do not die... THEY EVOLVE!'".
Jean Grey (TRN240) - Jean Grey from the past, seeing the current state of the world during Avengers vs. X-Men, her latent psychic ability awakened, making her a powerful Omega Class Mutant, even lacking the Phoenix's unique abilities. She is currently residing in the Jean Grey School with her other team mates with the exception of their version of Angel.
Franklin Richards (10235) - Unlike the above Jean Grey, this Franklin Richards instead came from the future to prevent the destruction of the world. Seemingly possessing the same abilities as his 616 counterpart, he is now a member of the Future Foundation, sharing Franklin's 5 designation.