Hand Off

Mike is introduced to Bill for the first time. Bill extends his hand for a handshake, but Mike flatly refuses the offer.

Can be done in a variety of situations for a variety of reasons. Mike may be an Ineffectual Loner or Anti-Hero who is merely displaying his Badass demeanor or desire to work alone. Bill may be some sort of villain or corrupt figure, and Mike is merely expressing his distrust or disrespect towards Bill by refusing his handshake.

A comedy version of this trope has Mike refusing the handshake for a humorous reason, such as Bill having just sneezed into his hand.

Anime

 * Inverted in Mobile Suit Gundam Wing: when future rivals Heero and Zechs meet face-to-face for the first time, Zechs offers his hand. After a pause, Heero accepts the gesture while remarking that he's never shaken hands with anybody before. This could be because Zechs has shown himself to be very honest and honorable, especially for a member of OZ.

Comic Books

 * Batman likes doing this. A lot. Even as Bruce Wayne.
 * One Dilbert etiquette book has Dogbert explain that this is the "right" way to respond if someone tries to shake your hand after you sneeze into it. The "wrong" way is to shake anyway and let them know afterwards, and the "playful" way is to shake and claim that the moisture is because you just washed your hand.

Film

 * In the 1998 film Godzilla, Matthew Broderick refuses to shake a nerdy scientist's hand after he just sneezed into it.
 * And it happens again in the first episode of Godzilla the Series.
 * Jim Hawkins does this to Long John Silver in Treasure Planet. At first, it's because he's a cyborg and happens to be offering a collection of knives instead of his hand in hand mode. Then it's just a sign of distrust.
 * In the film Mr. Brooks, Kevin Costner's character does this to Dane Cook's character.
 * In Blue Streak, Martin Lawrence's character, while posing as a cop, is introduced to a high ranking police officer who extends his hand to shake. He had previously spied the man applying lotion to his private parts, so instead hugs him.
 * Seem to recall this happening to Daniel Jackson in Stargate the movie, kinda a mixture of comedy of Jackson sneezing in his hand and contempt for him being in a military base in the first place.
 * In Tombstone, Doc Holliday refuses to shake the hand of the corrupt fop of a sheriff who is in league with the cowboy gangs.
 * Though this is because he's a Lawman, and has nothing to do with his corrupt foppery.
 * At the end of the OOO / Den-O All Riders Let's Go Kamen Rider, Eiji / OOO and Kotaro / New Den-O shake hands. Momotaros offers Ankh a handshake...DENIED.

Literature
"I thought the commander would offer me his hand, to seal our agreement. He did nothing of the sort. I regretted that."
 * In Robert Asprin's Another Fine Myth, Aahz offers to take protagonist Skeeve on as his apprentice, since Skeeve's teacher has just been murdered. Pleased, Skeeve offers his hand to seal the deal...and Aahz refuses, because he doesn't shake hands with apprentices. Skeeve wonders if he should have thought this arrangement through a little more.
 * Much later, while searching for a missing Aahz in his home dimension, Skeeve flinches at shaking hands with an apprentice. When he explains where he picked up the habit, Aahz's old magic teacher remembers him ("It was one of his less objectionable traits") and is able to help Skeeve look for him.
 * In Harry Potter, Harry refuses Malfoy's handshake when they first meet on the Hogwarts Express, rejecting him in favor of his future best friend Ron.
 * Twenty Thousand Leagues Under the Sea: The Professor Aronnax gets this treatment with Captain Nemo. Nemo doesn't distrust Wide Eyed Idealist Aronnax, is just one of the first of many signs to show how far Nemo has been Madden Into Misanthropy:

Live Action TV

 * In season five of 24, Jack Bauer reacts this way to his ex-wife's new boyfriend.
 * Monk is a germophobe and refuses to shake hands with just about anyone. If he's forced to, he will immediately turn to his assistant for a wipe.
 * In one episode he voluntarily shook hands as a sign of friendship with an ex-colleague of his, who was accused (even by Monk) of being in the drug-business (which he wasn't, which was proven of course.)
 * In another episode he shook hands with a succession of people, after the last one he immediately turned to his assistant for a wipe. The problem being, the last handshakee was black, leading to much accusations of racism.
 * In a Seinfeld episode, Elaine's boss has a cold. Having left his handkerchief in Elaine's office, he sneezes into his hand and is unable to shake the hand of a Japanese businessman when he sneezes, explaining that he can't because of the germs. The Japanese men assume he is implying that they might be diseased and cancel their business dealings.
 * The Mayor from Buffy the Vampire Slayer is portrayed as a neat-freak, and refuses to shake anyone's hand.
 * Victor Kennedy in the Doctor Who episode "Love and Monsters", allegedly because of his "eczeema"; it turns out.
 * Additionally, near the end of the second series, the Doctor is distracted by the presence of the Void Ship and fails to shake hands with the technician when offered.
 * Vince MacMahon on at least one occasion said "I don't shake hands with lawyers" on WWE.
 * On Angel, Cordelia introduces herself to the friendly but corrupt lawyer Eve. When Eve begins to introduce herself in turn, Cordelia interrupts her, saying, "I didn't ask."
 * Don does this to Pete Campbell in the pilot of Mad Men; it's fairly awesome.
 * House is not fond of physical contact and often will refuse handshakes, regardless of who they're from.
 * There was an episode of M*A*S*H whose B-plot involved Winchester's dental woes coupled with his fear of dentists. In the end of the episode, Winchester is about to shake the hand of the man who Hawkeye and BJ brought on to handle the episode's A-plot only for them to reveal he's a dentist; Winchester immediately retracts his hand. To be fair, this was more due to his fear of dentists than any attempt at being insulting.
 * In the original Ellen Degeneres sitcom Ellen, one episode involved the filming of a talkshow, which ended with the host shaking hands with the guests - onc the cameras and music stopped, so did he.
 * In a very very old episode of Baywatch a returning lifeguard refuses Mitch (David Hasselhoff's) handshake. However rather than being rude it's the first clue that an eye-disease is destroying his periphial vision.

Newspaper Comics

 * Dogbert's Rules of Etiquette has a page on how to handle handshakes. Dilbert denying a handshake because he just sneezed is labeled "polite"; another character snarking after shaking with a sneezed hand is "rude"; Dogbert lying and saying it's wet because he just washed it is labeled "playful".

Video Games

 * In Metal Gear Solid 2, Peter Stillman reacts this way upon being introduced to Iroquois Pliskin,.
 * At the end of Metal Gear Solid 3, the U.S. President awards the Distinguished Service Cross to Naked Snake. He shakes LBJ's hand, but.
 * Parodied in the Secret Theater skit, "Payback", where the DCI tracks down Naked Snake and bests him in a fight just to get that handshake.
 * In Final Fantasy VIII, Squall does this quite a bit, most notably to Zell and to the leader of La Résistance.
 * Everyone does this to Zell. It's one of the game's Running Gags
 * The player can do this at the start of Strange Journey. This earns you Chaotic points on the alignment meter (agreeing to shake hands earns Lawful points).
 * In the Mass Effect 2 DLC, Stolen Memories, you can offer to shake the hand of mercenary group owner Donovan Hock. He will refuse, because he's an all-around asshole.
 * In Mass Effect 3, Shepard offers Last of His Kind Javik a handshake during their first scene together, but he turns it down. In your last scene together before the final battle, he holds out his hand, and you can either take the Paragon Interrupt or turn him down.
 * In Mega Man Zero 2, Elpizo introduces himself to Zero as the new commander of the resistance and offers his hand after announcing his plans to destroy Neo Arcadia. Zero refuses both the plans and the handshake.

Webcomics

 * Questionable Content: On being introduced to Dave, Steve refuses to shake his hand because immediately preceding the introduction, he heard Dave watching porn. Yeaaah.