This motorcycle was owned by Scott Summers, and was used by both himself and Wolverine.
History
X-Men
The motorcycle, along with an assortment of other vehicles, was stored in the garage of the X-Mansion.
When Rogue left the X-Mansion following Mystique's ruse, Logan borrowed the motorcycle and left for Salem Center's train station once it was discovered she was there.
After Xavier informed Logan that Alkali Lake in Canada might hold some clues regarding his past, Logan again takes Cyclops' motorcycle for the trip.
X2: X-Men United: The Movie Prequel: Wolverine
Four days after leaving the X-Mansion, Logan reaches Alberta, Canada. During a gas stop to refuel the motorcycle, he encounters Sabretooth, who had been tracking him for days. The two soon engage in combat.
Just before Wolverine can kill his enemy, he notices a dog tag on Sabretooth that's very similar to his own. Questioning Sabretooth on the matter, the latter responds that he has always possessed it. In order to find further answers, Wolverine offers to buy Sabretooth a few drinks at a local bar.
Before they can continue their conversation, special forces break into the bar and confront the two mutants. Although their original intention was to apprehend Sabretooth, the soldiers quickly refocus their efforts on both men.
Wolverine and Sabretooth work together to fend off the attacking forces, but are ultimately subdued and ordered to be taken to an undisclosed covert ops base run by William Stryker.
A while later, both wake up in the cargo-hold of an aircraft en route to the base. As Logan attempts to escape by ripping through the fuselage, Sabretooth grabs his neck and throws him off, the latter crashing into a frozen river.
Logan survives the crash, and eventually makes his way back to the motorcycle to continue his journey to Alkali Lake.
X2: X-Men United
Logan eventually reaches Alkali Lake, but finds very little there to help him piece together his past. Disappointed, he returns to the X-Mansion.
Once Logan returns, he tosses Cyclops the keys, saying that his motorcycle was out of gas. Slightly annoyed, Cyclops tosses the keys back, telling Logan to fill up the gas tank.
X-Men: The Last Stand
Scott, grieving over Jean Grey's death, feels compelled to return to Alkali Lake - the site of her apparent death. He uses his motorcycle in order to get there. Once he arrives, he discovers that Jean was in fact alive - protected by a psychic cocoon. During a passionate kiss, Jean's Dark Phoenix personality surfaces and disintegrates his body.
Detecting Dark Phoenix's presence, Professor X sends Storm and Wolverine to Alkali Lake to investigate. There, they find Jean unconscious and bring her back to the X-Mansion.
It can be assumed that Cyclops' motorcycle was also brought back, as Logan uses a motorcycle to reach the Forest, making a final attempt to bring Jean back to the X-Men.