Spoiler started out having a white moped type thing which she used in her first appearance, Robin # 5, and Robin # 26.
However, something appears to have happened to it between then (perhaps she had to sell it after the quake?), as in Robin # 94, she was stuck using her pink mountain bike to trail her dad.
NOTE: In Detective Comics # 648, Stephanie was shown wearing what appeared to be a work shirt labeled "Burger Heaven". This may indicate that she was possibly employed at a fast-food restaurant. Later on, there was no evidence that she was employed. Perhaps she was able to afford the motorcycle while she was working, but for whatever reason, stopped working there and could no longer pay for insurance and gas for the motorcycle.
Batman then provided her with a new motorbike between issues. Unfortunately, the tires got wrecked when she tried to get into the Batcave in Robin # 98 (Bruce had sealed it off during the "Bruce Wayne: Murderer" arc without telling Steph).
It was never seen again after that, though mentioned in Robin #99'
("Do you have your bike with you?" "In the garage.")
When she was Robin, Steph rode in the passenger seat of the Batmobile, or the Batplane, despite her pleadings that she wanted to drive both of them.
Spoiler shows up in Robin # 174 using a motorcycle, looking similar to the one she had before.
Her primary method of transportation is usually jumping around rooftops with a grappling hook.
Recently, in Batgirl (v3) # 7 , Steph acquired a new vehicle as Batgirl. Oracle gave it to her to pursue a captured Batman. Oracle kept in her private garage under her apartment a vehicle similar to the Batpod from The Dark Knight film- only it was colored eggplant for Steph. Oracle told Steph it was "real fast" and warned her, "don't scratch the paint." Stephanie took to the pod right away, roaring down Gotham streets in it.
When Steph asked if there were "any guns on this boat," Oracle said no, so Stephanie improvised by using the vehicle's ability to eject her several feet into the air to go after her villain. The vehicle has a computer console inside that allows Steph to type in such commands. The vehicle is apparently called the Ricochet, because before ejecting Steph it said, "Now. Exiting. The. Ricochet."