The SHRA has been sketchy for quite some time; even years after CIVIL WAR, Marvel has never had a line wide simply definition of it, allowing writers to change details of it at a whim.
A Mutant Registration Act, requiring mutants to have to certify themselves as mutants to the government, ultimately was passed in Marvel's Congress, although enforcement of it has waxed and waned. During the "ACTS OF VENGEANCE" crossover storyline during the 80's into the 90's (where Loki manipulated various supervillains, including Kingpin, Mandarin, Wizard, Red Skull, Magneto, and a Doombot posing as Dr. Doom to hire super-criminals to attack heroes they don't usually battle, thus forcing heroes to fight unfamiliar foes), the U.S. government and the media began turning against superheroes in general due to the massive property damage and risks to lives that were taking place during this uptick of super-villain violence. I think Congress was debating something akin to a Super Powers Registration Act (or something similar) to force all superheroes to register to the government and act only under their behest. Many superheroes were opposed to this, especially Wonder Man, Wasp, and Mr. Fantastic. I believe the debate ended when Mr. Fantastic noted that not all superheroes have powers, they merely may be exceptionally skilled compared to a "normal" person, and used a device to determine that according to any measure to judge and force someone to register based on "super-potential", many within the Congress and the gallery would be considered super-beings because of advanced skills or intelligence.
You may be asking, "then why about 15 years later did all of those heroes almost blindly support Iron Man's brutal and overreaching enforcement of the SHRA, itself similar to the Super Powes Act"? The answer is because Mark Millar wanted a superhero fight, and didn't give enough of a damn to make it character accurate.
Basically, the SHRA supposedly requires anyone with super-powers or enhanced skills (such as enhanced combat skill or intellect) who engages or wishes to engage in crime-fighting to register with the government, reveal their identity to the government, be trained at a facility, and then possibly be dispatched to an officially sanctioned team in one of the 50 states (or be placed on "reserve", or dismissed if they do not pan out of the training). There are fuzzy areas. During CIVIL WAR, JMS and Millar appeared to contradict each other about whether the SHRA involved revealing your identity to the government and NOT the public, or to the government AND the public. Supposedly, Iron Man and Spider-Man chose to reveal their identities to the public as a show of support, although it may not have been technically required. After all, there are many things that are of knowledge to the government that are not officially told to the public or the media, although leaks (often politically motivated) happen. The anti-SHRA argument was of course that even with non-public reveals, hacking government files was akin to a commercial break for many villains, not to mention corrupt officials. In Marvel history, many menaces were created in government attempts to create superhumans, and there have been many politicians, including many then-current senators, occasionally involved in criminal enterprises such as The Corporation, or even being Skrulls. It also was seen as a metahuman draft; you could not REFUSE to register, lest you be thrown in prison or stripped of your powers via nanobot injections, dubbed SPIN-Tech. When Rage was proving overly combative and violent in his Initiative training, Gauntlet seemed to suggest his powers be removed until he was chronologically 18, although this order was either never put through or not enforced (or, after suffering a coma-inducing beating by Slapstick and being revived prematurely twice, once via drugs and another time from his alien gauntlet, he may have suffered some memory loss).
In THE LAST DEFENDERS, it was revealed that there is supposedly a part of the SHRA that claims that if a registered superhero commits an improper act while under official duties, they can be stripped of their superhero identity and not legally allowed to act under that name again. Kyle Richmond was forced to "surrender" his identity of Nighthawk after he improperly uses SHIELD resources and technology to rescue a missing SHIELD agent Joaquin Pennysworth (son of an old business friend), then essentially hiring Atlas, Junta, and Paladin as "mercenaries" for a mission not officially sanctioned by Stark and performed over the heads of the Mighty Avengers, who arrived after the situation (a battle against U-Man and Roxxon pirates I believe) was resolved. Kyle wasn't stripped of whatever powers he had, but did have to surrender his armored flying costume and was forbidden to operate as a superhero (I believe Richmond at this time still gained superhuman prowess at night, although he also relied heavily on his tailored Nighthawk costume). Kyle would later allow Joaquin to operate as Nighthawk and founded a Defenders team not officially sanctioned by the Initiative as a "private enterprise", but who knows where this wound up.
The SHRA was officially passed by Congress during CIVIL WAR, and hence that was why it caused such a stir; Iron Man proposed and carried out strict and often overzealous enforcement, arresting superheroes who even had little or no power just for refusing to sign on, such as Prowler (arrested for basically leaving his apartment in costume). Captain America likely did not help matters by founding a resistance of unregistered heroes that including the mass murderer, Punisher. But honestly CIVIL WAR was all over the place, obviously had rewrites even if everyone denies it, and was more about fights and moments than a coherant story. After CIVIL WAR, though, Iron Man scaled back his zealous enforcement of it, although unregistered superheroes were still violating the law and subject to arrest, such as the New Warriors team led by Donyell "Bandit" Taylor and filled by former, depowered X-Men (which has since apparently disbanded). Counter-Force was a team led by Justice and filled with technically registered superheroes, all former New Warriors, that strove to defend against Initiative corruption while not technically violating the SHRA, supposedly. They have since changed their name back to New Warriors and are working with the presumably unregistered Donyell Taylor, and Osborn seeks to officially arrest them since taking over. Stark also "allowed" the New Avengers, another team of unregistered or anti-registration heroes, to "escape" several battles for plot convenience.
Punishment for violating the SHRA varies from either a forced depowerment in some stories or being thrown into Prison 42 in the Negative Zone, a hellish prison where the dimension itself could induce pain or psychosis. 42 has seen been breached and taken over by Blastaar, with any inmates who didn't join him having presumably been slain. Anyone the THUNDERBOLTS mangled or arrested wound up in 42; so far only Jack Flag, who joined the Guardians of the Galaxy, has managed to escape 42 without joining the Thunderbolts or being slain by Blastaar. Many heroes arrested and placed there during CIVIL WAR were freed by Captain America, with some gaining pardons and officially registering after he surrendered to authorities (and was killed). The heroes arrested after that, though, who knows. Punishment for violating SHRA tactics despite being officially registered varies from being arrested and thrown into 42 or a similar prison, being depowered, or stripped of your superhero identity and "forced" into retirement.
Retirement is also supposedly a way out of the SHRA; if you have super-powers but refuse to act as a hero, basically retiring, you do not have to register if you seek to remain inactive. Angelica Jones, a.k.a. Firestar, avoided the Initiative by retiring in FRONTLINE #2, and remains so in the pages of NOVA. I would imagine the government would still need to know a hero is retired to not go after that and that may involve some loss of privacy, although on the other hand being "retired" would make it hard to be arrested for violation of said act, since it is hard to find a hero with a secret identity who no longer wears the spandex.
The Mutant Registration Act was supposedly lumped into this as the X-Men were included in the SHRA, although aside for Bishop, remained neutral. Colossus briefly registered and was active in Stark's gathering of Defenders stationed in New Jersey, but after a battle with the Sons of the Serpent resulted in a trashed Atlantic City due to poor teamwork, that team was scuttled and Colossus apparently sent back without any demerits. One could be disturbed by the idea of an act that was targeted against mutants has been "expanded" to target not only any superhuman regardless of origin, but any superhero who happens to be considered above average in skills, intelligence, or manages to use some advanced weapon or armor.
One supposed the SHRA could be challenged as unconstitutional in the Supreme Court, but that seems unlikely; Marvel's government has been infamously corrupt for decades of stories (deputizing criminals for various reasons many times) and the public usually remains paranoid and resentful towards most noble heroes, only championing the obviously corrupt (Osborn, the Thunderbolts).
