You can do the Free Run mode in Practice or go to the Arcade and do a time trial. Every time you complete a fast lap it will be placed on a leaderboard for that track on the screen where you can start your drive, tune your car, driving options, and exit back the main menu. These are only your times as there is no online lap leaderboard.
Also, if your fast lap with a car is slower than the 10th fastest lap on that track it will not show up. You can always delete the hot lap data by clicking on the button next to the exit button while in either mode.