First, There Was Ramming Speed
By 1210 BC, the first recorded naval battle was fought between Suppiluliuma II, king of the Hittites, and Cyprus. Before crossbows and black powder had been invented by China, early navies relied on ramming to sink enemy vessels. Explosive ramming (with spar torpedo) was not used successfully until the H. L. Hunley sank the USS Housatonic in 1864.
Human oar power with slaves or volunteers limited ramming speeds to the single digits (about 4 mph for the semi-submerged Hunley).
Conventional Submerged Speed from Wikipedia rumor source
Due to the stealth nature of miltary vessels the following rumors are unconfirmed and most certainly are somewhat suspect:
The Alfa class submarine, reportedly attained short speed bursts of 44.7 knots (51 mph) while submerged. The British Spearfish torpedo designed to counter high speed Russian submarines, such as the Alfa class, is rumored to have a speed in excess of 70 knots (80 mph).
Cramming (In Bubbles) Speed ibid. Wiki source
The Russian rocket-powered supercavitating torpedo VA-111 Shkval is reportedly capable of speeds in excess of 200 knots (230 mph). The US Defense Advanced Research Project Agency (DARPA) began the Underwater Express program in 2005 to research applications of supercavitation. DARPA awarded contracts under the program to General Dynamics and Northrop Grumman in 2006. Research is now taking place at the Naval Undersea Warfare Center (Newport, Rhode Island, USA) and Space and Naval Warfare Systems Center (San Diego, California, USA).
Warp Speed (Not Available)
In the fictional universe of Star Trek, the warp drive is faster-than-light propulsion portrayed as propelling spacecraft to many multiples the speed of light (about 299,792,458 m/s).