Walking, talking, purring class. She doesn't steal; she liberates. She doesn't lie; she fibs. She steals from the rich and keeps it all for herself, but she's so graceful and easy on the eyes that you'll let it slide.
When The Hero first meets her, on a rooftop or a darkened bedroom, he'll be paralyzed with indecision — should he arrest her or ask for her number? This dilemma is usually short, as she takes advantage of his distraction to escape/knock him out.
Bonus points if she has a cat theme, is named "Katherine," "Kitty," or "Felicia," or makes bad feline puns.
She doesn't always wear a Spy Catsuit, but if she does she wears it well.
Either way, Classy Cat Burglars are sophisticated and highly skilled. They target only the finest items (and best-protected) for "liberation", and pride themselves on leaving little or no clue on how they accomplished their burglaries. A fair number of them are independently wealthy and couldn't care less about the money; they just want a good challenge. (Genre Savvy detectives understand that a well-guarded, priceless item is the best bait in the world for these sleek critters). The actual term "cat burglar" comes from the notion that such a person is quiet as a cat (one that never claws its way up the drapes, gets into fights with the dog and tears your refrigerator to bits).
Like her spear counterpart the Gentleman Thief, this felonious feline usually regards the police with a certain amount of disdain and condescension, and frequently leaves behind "calling cards" announcing who performed the crime. With a Worthy Opponent, she may have a less adversarial relationship, verging at times on friendship (and if the opponent is of the opposite gender, fraught with UST).
Compare Phantom Thief. If the hero is very, very (un?)lucky, may result in Dating Catwoman.
Train Heartnet from the anime Black Cat is an assassin and a rare male example of a criminal with a cat theme. The series title is his alias and he shamelessly sports a fluttering black cloak and a bell on a red string as a necklace.
Also Rinslet, who doesn't have a cat theme but definitely fits all of the main requirements.
The very efficient thief and Master of Disguise Masami Hirota (or better said, Akemi Miyano) in Detective Conan is very beautiful and looks really pretty whether dressed in red or in white, but she's less traditionally fanservicey than many examples of this trope. Also unlike others, she didn't act on her own will, but as a member of a local evil organization. And when she tried to leave... they killed her.
Catwoman is perhaps the ultimate incarnation of this trope.
Felicia Hardy, Marvel's Black Cat, also plays up this theme.
Keira the Thief in the Dragaera novels; she's one of the most talented thieves in the Jhereg crime syndicate, to the point where any unsolved burglaries are usually attributed to her, and she's polite and classy when speaking to others. She also has no need of wealth, as she is the immortal sorceress Sethra Lavode, who took up the guise in order to gain intelligence on the syndicate and kept it up when she discovered that committing the thefts without using her powers gave her the first real challenge she'd had in tens of thousands of years.
Sandra Paris, a.k.a. the White Queen, Nick's sometimes rival in the Nick Velvet short stories of Edward D. Hoch.
Kat from Heist Society by Ally Carter is actually named after this trope. The author was inspired by the phrase and decided to write a book about a burglar that was actually a "Kat." Kat mainly steals expensive paintings that are under high security. She also has that moment where all the male characters get distracted from their heist because she looks so drop dead gorgeous.
Though her cousin, Gabrielle, probably plays this trope straighter than Kat.
The In Death series: Magdalana is presented as this in Innocent in Death. However, it gets subverted when it's revealed that she is a thief and not as classy as she pretends to be, with Roarke pretty much saying so to her face.
In Void Moon Cassie fits this role, but is a slightly more realistic version of this character in that she doesn't pull impossible heists, instead focusing on much smaller targets. She and her partner robbed high rollers at a casino before being caught and having her partner killed.
Libby Craddock from Texas Kidnappers by J.T. Edson. Her exact level of classiness depends upon her mood.
Tracy in Sidney Sheldon's If Tomorrow Comes becomes one after overcoming a Trauma Conga Line that leaves her unable to find legitimate work. She works for an employer who assigns her to targets that range from Jerkass to outright corrupt, and besides the usual cat burglar hijinks uses her talents as a Master of Disguise to snooker unsuspecting targets, to the point that Interpol thinks her work is that of an all-girl gang. Her romantic interest? A rival Gentleman Thief!
Vala Mal Doran of Stargate SG-1 was something of a failed cat burglar. In addition to failing to steal the Prometheus and a Goa'uld Naquadah bomb, she fails to seduce Daniel Jackson. Repeatedly. And Lt. Col. Mitchell. And Teal'c. But mostly Daniel. Even getting to the point of possibly stealing a gigantic well-defended armed and heavily populated ship single-handedly is pretty damn cool. If she'd chosen anyone besides Daniel to tie up she probably would have made it. Plus, as she points out, the Goa'uld Naquadah bomb was inside alien technology she'd never seen before and...well...she rushed it. And she did eventually seduce Daniel, and discovered he was resisting before because he thought she wasn't as interested in him as he was in her. Plus she's played by Claudia Black, so she's gorgeous.
Subverted in Firefly, when Saffron tries this act on Mal. It fails.
The "Taffy" imprint in the Dollhouse episode "Gray Hour," first in Echo and then in Sierra (who helps rescue Echo after she gets mindwiped by cell phone).
Sandra from Legend of Mana presents herself as a classy catburglar specializing in jewel theft, but the truth is much darker. The Jumi race's "cores" are gemstones, and Sandra is collecting Jumi cores. She doesn't care if the Jumi who's core she's trying to steal is still alive, either.
Kay Faraday, self-proclaimed Second Yatagarasu in Ace Attorney Investigations, is going for this. Given that she's an excitable teenager who doesn't even know the truth about the first Yatagarasu, she's got a ways to go.
The Thief units from Disgaea are a much more brightly coloured than usual version of this. Their clothing consists of a cross between a brightly colouredBlatant Burglar getup and a Cat Girlcostume (knee high striped socks, cat eyed goggles instead of a mask, cat pawed boots and gloves, etc).
"Mind if I steal your heart? ♥"
Monster Hunter has Melynxes. Though not so much classy as cute, but they even have their own "cat's paw" weapon to steal your stuff with.
In Casey and Andy, it is eventually revealed that next door neighbor Jen is an internationally known jewel thief, and this is why Quantum Cop won't date her. It turns out that clues about this go back for years.
Shego in Kim Possible is generally seen as a thief when not being employed or being in vacation. In fact, the one episode Kim forgot about crime fighting due to amnesia, Shego spent the episode stealing stuff, instead of taking over the world stuff.