Shadowman - Under the skin, this game is pure evil. The serial killers and music freaked me out, Out of my 3 suggestions i'd easily recommend this one over the others. There's spoilers here but check the plot out - http://en.wikipedia.org/wiki/Shadow_Man_%28video_game%29
Love Shadowman but I can only play the Dreamcast version. The PS1 version is atrocious.
For me, there were great racers like Wipeout XL, Ridge Racer 4 and Need For Speed High Stakes and even Driver
Weird quirky games like Incredible Crisis and Looney Tunes Sheep Raider
Fun shooters like Medal of Honour/MOH Underground and Syphon Filter
Point and click games like Blazing Dragons, Discworld and Psychic Detectives
Horror games like Silent Hill, RE 1-3 and Vampire Hunter D
Shooters like Future Cop LAPD and Ghost in the Shell plus shmups like Einhander and even decent versions of Descent/Forsaken
3rd person action/stealth like MGS, Tenchu, Dino Crisis and Fear Effect
The PS1 had all the bases covered.