Flair Bartending

"It don't seem natural to me, throwing up a drink before you've even swallowed it."

—Reg Deadman, Goodnight Sweetheart, "Flash, Bang, Wallop"

Drink-making with an eye to showmanship, often incorporating juggling, dance moves, and nonstandard use of bar equipment. Possibly more prevalent in Real Life than in media; any given media portrayal has a good chance of being an homage to Cocktail.

Can also be done with food, particularly ice cream.

Anime and Manga

Bartender averts this, or plays this straight, depending on your point of view.

Film -- Animation

Long John Silver in Treasure Planet uses his multi-function cyborg arm to cook like this.

Film -- Live Action

Cocktail is the Trope Codifier, and may be the Ur Example of media portrayals as well.
Coyote Ugly.
Parodied in Jane Austen's Mafia!, when the priest mixes an Umbrella Drink for a distraught Joey.

Live Action TV

The King of Queens, episode "Pour Judgment." Doug takes work as a bartender to fulfill an old dream. Broken glass ensues.
How I Met Your Mother, episode "Three Days of Snow," when Ted and Barney briefly take over McLaren's.
NCIS's Abby was shown to have flair bartending skills.
One of the Challenges in Beauty and the Geek.
In A Bit of Fry and Laurie, Stephen Fry does this in the cocktail sequences that end all the season 3 and 4 episodes.
Carla did several stunts to win a best barmaid contest in Cheers, including a complete flip while holding a platter of drinks.
JD and Turk of Scrubs did this in one episode.
The Charmed episode "Feats of Clay" has Piper cast a confidence spell on a meek waiter so he'll have the confidence to ask a female coworker to marry him. Instead, he becomes a hyperconfident bartender, not very interested in his coworker anymore. Phoebe lampshades this by asking "I thought you were supposed to make him confident, not turn him into Tom Cruise?".
In the season five episode of House, "House Divided", the title character did this...in the hospital's morgue. Long story short, House needed to add "chemical burns" to a guy's cause of death.

Video Games

The Sims 3 Late Night has this as an entire skill.
Although the Citadel bartender in Mass Effect 2 pours drinks normally for the most part, he adds some style when Commander Shepard decides to down ryncol.

Web Original

In the Paradise web original fiction setting, a bartender who Changed to take on monkey characteristics bartends this way, using all five limbs. Because he looks perfectly human to people who can't see through the Change, the way he does what he does is perceived as puzzling to say the least.

Western Animation

Phineas and Ferb do this while making milkshakes in the episode "Out to Launch."

Real Life

Jerry Thomas, likely the Real Life Ur Example.