A type of sublight vessel was flown in the 2360s by such groups as Nova Squadron, who in 2368 attempted a, by then banned, five craft formation Kolvoord Starburst maneuver near Titan. (TNG: "The First Duty")
The Academy cadet ships were small, one-pilot, blunt-winged vessels, with a fairly flat cross-section; pilots had to wear flight helmets. The craft were equipped with landing struts, proximity alarms, and emergency transporters.
Warp-capable Federation attack fighter-type flight trainer schematics were part of an information download package aboard the USS Voyager. When One was assimilating as much information as he could, the flight trainer could be seen on a screen he was learning from. It was later seen when Seven used a new device on her Borg alcove to upload information directly into her brain. (VOY: "Drone", "The Voyager Conspiracy")
Background information Edit
Two versions of the Academy flight trainer have appeared on screen. The actual, original ships seen in "The First Duty" and the side view of the Federation attack fighter, first (and only) seen as an okudagram in "Drone". The side view created by Doug Drexler, was first used in the Star Trek Encyclopedia (2nd ed.), in the article about the flight trainer craft.
The Academy's space force consisted of at least three academy training wings headed by wing commanders. The wings were divided further into squadrons commanded by squadron leaders. For example, the Third Academy Training Wing was commanded by Captain David Livingston. The Falcon Squadron, commanded by Captain Brad Yacobian, was assigned to this training wing. The operations of these wings were managed by Academy Flight Ops. Pilots who successfully completed their training were awarded the Flight Proficiency Award. 
The original flight trainer, merely described as "five small SHIPS of the Nova Squadron" in scene 18 of the script, and simply referred to as a "Trainer craft" by its designer, was designed and signed off on in January 1992 by Senior Illustrator Rick Sternbach.  A studio model was actually built by Gregory Jein but was only seen in two brief scenes and used only in "The First Duty". "Greg Jein built a physical miniature about 14" in length. There was no major CG version that I know of aside from five tiny diagramatic ships (...)", Sternbach remarked.  The otherwise indiscernible symbol in front of the cockpit was that of Cobra Commander from the G.I. Joe: A Real American Hero franchise, which Jein, a fan of the franchise, added as a subtle homage. wbm
A copy of the model, made by Jein for Sternbach and painted and detailed by the latter, along with five of Sternbach's sketches of the design, was later offered as Lot 298 in the The Ultimate Sci-Fi Auction of 26 April 2003, estimated at US$800-$1,200, where it sold for US$900.
The original studio model itself, having escaped the 40 Years of Star Trek: The Collection and It's A Wrap! sale and auctions, is still in the possession of CBS Consumer Products and has been on tour as a display piece, starting with the Star Trek: The Adventure tour , and subsequently appearing in the Star Trek The Exhibition tour, as late as 2012.  Worn from years on the tours, the model was by CBS remitted into the care of ILM modeler John Goodson for some much needed restoration in 2015. 
The Federation attack fighter standing in for the trainer in the Encyclopedia may well have stemmed from the fact that the flight trainer was mostly depicted as a computer graphic in the episode and that it was either forgotten, or not realized by the authors due to its very brief appearance that a model had actually been built. "There was some editorial confusion over which little ship was the trainer," Sternbach conceded in 2013.