From The TimeSplitters Wiki

The Challenges are similar to Arcade League, however in TimeSplitters 2 and TimeSplitters: Future Perfect they cannot be recreated in Arcade Mode. TimeSplitters 1's Challenge Mode features both unique challenges and missions that can be recreated in Arcade Mode. Many of the challenges have a minigame feel to them and objectives are often more absurd than those in the League.

Completing these challanges not only unlocks more challanges but additional characters and cheats for use in both arcade and single player.

TimeSplitters 1-A: Behead The Undead 1-B: Putrid Punchout 1-C: Dusk of the Dead 2-A: Flock Around The Dock 2-B: Dock Duck Shoot 2-C: Crispy Duck 3-A: Tin Man Trial 3-B: Lobster Run 3-C: Bowl them Over 4-A: Shame if Something got Broken 4-B: Don't Wait Around 4-C: Brick Flung High 5-A: First Impressions 5-B: Lasting Impressions 5-C: Who's he Trying to Impress? 6-A: Barrel Blast 6-B: Bodyguard 6-C: Heist 7-A: Shop 'Til you Drop 7-B: Sorry, Was That Your Bag? 7-C: Everyone Must Go! 8-A: Girls 'n' Boys 8-B: Dinner Dates 8-C: Bone Grab 9-A: I Can't Hear Anyone Screaming 9-B: Flight Delay 9-C: Space Vandals

TimeSplitters 2 Glass Smash Pane in the Neck Bricking it Stain Removal Behead The Undead Fight Off The Living Dead Sergio's Last Stand Day of the Dammed Infiltration Silent but Deadly Trouble at the Docks Escape from NeoTokyo Banana Chomp Gone Bananas Monkey Business Playing with Fire Cut-out Shoot-out Take 'em Down Fall Out Pick Yer Piece Timesplitters 'Story' Classic Badass Buspass Impasse But Where Do The Batteries Go? Hit Me Baby One Morgue Time Monkeying Around Simian Shootout Monkey Mayhem Dam Bursters

TimeSplitters: Future Perfect Behead The Undead Brain Drain Rare or Well Done? Boxing Clever Cut-out Shoot-out Hart Attack Come Hell Or High Water Balls of Steel Cat Driving The Cat's out of the Bag Lap It Up The Cat's Pajamas Super Smashing Great Avec Le Brique Absolutely Potty Don't Lose Your Bottle Timesplitters 'Story' Classic Queen of Harts Sammy Hammy Namby Pamby Glimpse of Stocking Monkeying Around Electro Chimp Discomatic Melon Heist Brass Monkeys Miscellaneous Challenges Cortez Can't Jump! TSUG: Timesplitters Underground Plainly Off His Rocker