(written from a Production point of view)
In the history of Star Trek, there have been numerous visual effects used to portray warp drive.
Star Trek: The Original Series Edit
Originally, Gene Roddenberry wanted the effect of the USS Enterprise engaging warp drive to cause the starship to become transparent. This effect was abandoned, however. (The Star Trek Compendium, 4th ed., p. 38)
TOS movies Edit
Doug Drexler noted, "I never liked what they did in the movies with the flashy red, white, and blue. I hated that." ("The Next Generation: Back to the Future", The Fifty-Year Mission: The Next 25 Years)
Star Trek: The Next Generation Edit
Doug Drexler enthused, "I love the rubber-band snap [....] I thought the rubber-band thing was very clever." ("The Next Generation: Back to the Future", The Fifty-Year Mission: The Next 25 Years)
Star Trek: Enterprise Edit
Star Trek Nemesis Edit
In Star Trek Nemesis, a trail of smoke briefly appears behind the nacelles of the USS Enterprise-E when the ship accelerates to warp speed. The effect has not been used in any other Star Trek production. 
Star Trek Into Darkness Edit
Conceptually, the Enterprise's warp contrails were intended to be a new addition made to the ship between Star Trek and Star Trek Into Darkness. Regarding the addition of the contrails, Lindelof explained, "That was driven by a few things. One was to show the audience that Scotty has made modifications to the Enterprise between the two films [...] since Scotty is now the [chief] engineer." The filmmakers were mindful that, in adding the warp contrails, they didn't deviate from canon too much, such as if they had added a cloaking device to the Enterprise's systems instead. The warp contrails were depicted with CGI. Said Lindelof, "Roger Guyett, our visual effects supervisor, took J.J.'s phrase 'warp contrails' and ran with it [....] Roger tried a few tests, some of which felt a bit too radical, but it was basically us [...] saying 'wow, that looks cool.' And we went with it."