stra-tek:

A short list of magical transporter abilities

Feel free to add ones I’ve forgotten and attack/defend the plausability/implausibility of them.

Transwarp Beaming

From a relatively stationary start to a ship at warp. From Earth all the way to Kronos (144 light years). A software patch to the regular transporter, invented by Scotty, whose alternate self almost died using it to beam to the Enterprise, and who had previously lost Admiral Archer’s beagle when attempting to demonstrate it (Star Trek, Star Trek Into Darkness)

Subspace Beaming

Across several light-years. A minor modification considered dangerous, known to the Federation and Ferengi. Possibly the same as transwarp beaming. (TNG: “Bloodlines”) The Dominion also had transporters with extreme range (10+ light years), they could use theirs to pluck people from distant targets, something Transwarp beaming may be unable to do.

Beaming Into an Alternate Reality

First done accidently during an ion storm, replicated a century later using a handheld gizmo which interfaces with a regular transporter (downloading extra software?) (TOS: “Mirror, Mirror”, DS9: “Shattered Mirror” etc.)

Beaming Through Time

With the aid of some time-limited technobabbly particles, O’Brien was able to beam people to various periods in Earth’s history. (DS9: “Past Tense”)

Cure All/Anti Ageing/Re-Ageing

Where to start? Kirk and Spock are de-aged in TAS: “The Lorelai Signal”, the entire crew are made their proper sizes again in “The Terratin Incident”, the baby crew are re-aged at the end of “The Counter-clock Incident”, Dr. Pulaski is cured of an ageing disease in TNG: “Unnatural Selection”, the crew are de-aged into children in "Rascals" and it’s reversed the same way at the end. I’m sure there are a few I’ve forgotten.

Epic Pause Mode

Scotty kept himself alive in the transporter for 70 years, in TNG: “Relics”. The guy he was in there with didn’t make it.

Splitting, Duplicating and Merging

Kirk is split into good and evil halves in TOS: “The Enemy Within” but both are merged again at the end, Captain Christopher is (pointlessly?) beamed into his slightly-earlier self in “Tomorrow is Yesterday”, Will Riker is xeroxed in TNG: “Second Chances”, Tuvok and Neelix, with the aid of a magical plant, merged into one being and then somehow separated again months(?) later (VOY: “Tuvix”)

Become a ghost

The inventor of the transporter lost his son in early experiments (ENT: “Daedalus”), and he spent the next 20 years appearing as a ghost and attacking people.

So basically, someone with a regular transporter and the right set of software mods could beam anywhere in the galaxy, become virtually immortal, skip centuries in an instant, raise an army of clones and hop between universes, Sliders-style.