Controversy-Proof Image

Musicians from certain genres can survive scandalous controversies that would destroy or cripple other, pure mainstream, artists. For instance, any pop artist who said or did anything politically controversial, or something that were legally reprehensible and possibly career ending, could normally be survived by Alt/Rock, Heavy Metal, or Hardcore Hip Hop/Gangsta Rap artists. Pop artists come under scrutiny and they have to backpedal, clean up their image etc. With the other aforementioned genres, it probably helps their image. Sure, the Moral Guardians will whine real loudly, but nothing will ever come of it. The artist's image and fanbase remain intact.

This could also be true for actors and athletes who are probably charismatic and charming enough that the general public won't care about what ever scandal is plaguing them—if they don't rally behind them for it. A good example is probably David Letterman, as opposed to Tiger Woods. Whether or not this is fair is up for debate.

Compare No Such Thing as Bad Publicity. Contrast Convicted by Public Opinion (where it doesn't matter if someone even was guilty of something wrong, everyone seems to hate them anyways), Contractual Purity, and Overshadowed by Controversy. Somewhat related to The Tyson Zone, where celebrities get so bizarre that we stop being surprised about their latest escapades.

Examples of Controversy-Proof Image include:

Literature

Mentioned in Animorphs: When a Yeerk inspector comes to see how Visser Three is doing, they decide to run a major smear campaign, as this often works on humans "but not actors or politicians. They're like immune".

Live-Action TV

The black metal episode of Bones had Booth offering to charge the various metal band characters with assaulting a federal officer and such in exchange for information.
There was a similar bit in an NYPD Blue with a rapper who needed the "street cred."
The trope is played with in The Defenders court drama, coming to an aversion. A rapper is suspected of killing a rival musician, and it doesn't help his name is "Killer D." The next day while performing on stage, he raps about how he killed the guy. The police accept that as a confession and take him in.

Newspaper Comics

In The Boondocks, rapper Gangstalicious is charged with assault and possibly raps about it in a song called "Play it for the Jury", which the judge allows into evidence and is indeed played for the jury.

Professional Wrestling

CM Punk was like this in his early Indie wrestling days: his willingness to cross several lines (insulting audience members with below the belt comments, homophobic references, casual swearing, and other mean spirited insults and put-downs) in order to get the crowd to boo him didn't do much to hurt his standing as one of the top wrestlers in the country. Which in turn led to an incident in 2011 (when Punk's contract was about to expire and the WWE was deciding whether or not to renew it), when Punk yelled at a fan and called him homo and the video of the incident ended up on TMZ. The WWE and Punk quickly apologized and it didn't hurt Punk's image at all with fans or his standing with the WWE, who promptly resigned Punk to a lucrative new contract.
Brutally subverted with several wrestlers: Chris Benoit's reputation turned to mud after he killed his wife and son and then himself; similarly, Steve Austin's image never quite recovered when he was arrested for spousal abuse. The later claimed it was an incident born of 'roid rage, running before the cops arrived.

Video Games

The Telltale Games game Hector: Badge of Carnage has the main character bribe a street punk by offering to frame him for various crimes that will help his street cred. You end up stealing his pants and charging him with indecent exposure. He is quite happy since a sex crime is much better than a run-of-the-mill assault and battery.

Real Life

A definite example would have to be Black Metal. People like Varg Vikernes have had their images boosted (in the minds of fans) thanks to convictions of arson and even murder.
- Jaywalking, too?
Heather Locklear (the previous Mrs. Tommy Lee) publicly warned Pamela Anderson (the then-about-to-be Mrs. Tommy Lee) about Tommy Lee, pretty much on the basis that, as a bad boy rocker, Mr. Lee was controversy proof and therefore had nothing to lose by doing stuff that could damage Ms. Anderson's career. One sextape later, Ms. Locklear could say "I told you so", although it doesn't seem to have done Ms. Anderson's career that much harm.
Paris Hilton would appear to have only become more famous after the sort of revelation that put Rob Lowe into obscurity for a decade.
- The same thing can be said for Kim Kardashian.
R. Kelly was once a good example, as people previously thought his career was doomed thanks to his statutory rape/and child porn case. But then he released his album Chocolate Factory which became one of his biggest albums amidst the controversy - so much so the media stopped scrutinizing him when they realized he wasn't going to be another sensationalized Michael Jackson train wreck. This controversy-proof image did not last for long though, as mounting public pressure and the documentary series Surviving R. Kelly ultimately led to Kelly being dropped by his record label and later convicted of sex crimes involving minors.
The backlash against Bodycount for their song Cop Killer was initially No Such Thing as Bad Publicity, leading their debut album to go gold. Long-term, though, they quickly faded from the mainstream, although they have released several albums since.
- Former lead singer Ice-T now enjoys a second career as an actor, most famously on Law and Order SVU, where he plays... a police detective.
Rapper T.I. still has a very strong career despite all of his run-ins with the law, though his latest arrest is certainly taking a toll on his fanbase.
Child porn and other crimes overwhelmingly viewed as heinous by the public will kill the career of most. The few exceptions like Pete Townshend and Pee Wee Herman still carry the stigma of the accusation.
When Margaret Cho appeared on Dancing With the Stars, some tabloid ran an article about how the stars were scandalous backstage, and she was included. But since this is Margaret Cho we're talking about, the effect wasn't really the same.
Chris Brown is managing to pull his career back together after his arrest for assaulting then-girlfriend Rihanna and his sentence to five years probation. His image is not completely as clean as it was before, but he definitely still has a fanbase (with some fans even blaming Rihanna for the abuse).
Charlie Sheen has proven to be incredibly resilient to bad press and controversy. What would kill most other actors' careers dead just made Sheen a greater celebrity. To note: his first run-in with drug issues came in the '90s, when his father Martin Sheen had to hold a live press conference to beg his son to stop using. Many people thought he was burned-out...and then he came back with a vengeance on Two and A Half Men, where he essentially played a caricature of himself (and wound up become one of the highest-paid actors on television). Then he went on a tear by sleeping with hookers, hiring two porn stars to live with him as his "goddesses", took potshots at CBS and TAAHM creator Chuck Lorre (which got him fired) and generally went batshit crazy (so much so that he was a media punchline). The end result of that was CBS paying him an incredibly handsome sum of money for the unproduced episodes of his show, Sheen becoming an internet hero thanks to his first televised interview after his meltdown, praise from fellow celebrities and fans, and a lucrative deal with the FX Network to develop a series based on the 2003 film Anger Management. Apparently, it's good to be Charlie Sheen. This would finally end when it was shown he was HIV positive and intentionally spread the disease, causing his excommunication from production circles.
Mario Lopez has continued to host numerous TV shows (such as Extra) since his Saved by the Bell days ended even though he infamously cheated (if not well throughout the relationship) on his wife Ali Landry virtually a day after their wedding (the marriage was annulled shortly thereafter).
Tom Brady didn't suffer any effects to his squeaky-clean image after he cheated on his pregnant girlfriend and then left that girlfriend for the other woman. Some fans even congratulated him for it.
Sports announcer Mike Tirico is one of the top personalities on ESPN (calling Monday Night Football, the PGA Tour, the NBA, etc.) even though back in the early 1990s, he was suspended by ESPN for sexual harassment. In fact, Tirico's perverted tales have been documented in several "behind the scenes" books regarding ESPN.
The Dixie Chicks made the mistake of publicly announcing their distaste for the then-current president overseas. For reference, the Dixie Chicks are a country music act, and the president at the time was George W. Bush, who at the time had broad national appeal with the type of folks that listen to country music. On the other hand, Willie Nelson gets far less controversy for his political views (especially as a strong advocator for marijuana legalization) despite having many of the same views as the Dixie Chicks, primarily because of his "outlaw" image.
Salman Khan has been India's bhai (big bro) for over 30+ years now, and not even tweets supporting terrorist acts nor convictions for vehicular manslaughter and poaching have diminished his popularity. When it came to light that he harassed former girlfriend Aishwarya Rai, the only career stalled because of the scandal was the one of Vivek Oberoi, the whistleblower.
Rodrigo Duterte is arguably a polarising figure in the Philippines, yet despite intense criticism both in his home country and abroad for, among other things, his fiery, profanity-riddled rants (one such incident being a story he told about Pope Francis' visit in 2015 where frustrations over traffic due to said visit caused him to make a Precision F-Strike joke at the Pope's expense) and bombastic statements towards local and foreign statesmen, he still remains a popular figure amongst the masses due to his "tough guy" image. Not that there aren't any critics of him who are understandably sick and tired of his antics, though.