Ps 1&2.

I never really played those consoles much past childhood but these were my favourites:

PS1
WCW Mayhem
Simpsons Wrestling
WWF Warzone
Crash Bandicoot
Spyro

PS2
SmackDown vs Raw
Scooby Doo Night of 100 Frights
GTA Vice City
Sly 2: Band of Thieves
Showdown Legends of Wrestling

*PS1*
Tomba!
Street Fighter Alpha 3
Mortal Kombat Trilogy
Jonah Lomu Rugby
Spider-Man 2

*PS2*
GTA San Andreas
Capcom vs SNK 2
Blade II
Need For Speed: Underground 2
Tony Hawk's American Wasteland

Cool idea for a thread because the PS2 especially had some of the best games ever. In my opinion still the best console ever.

I'm not sure really on my favourite PS1's but I'll get back to them.

PS2
NFS Underground
NFS Most Wanted
FlatOut 2
Burnout 3
MotorStorm Arctic Edge

PlayStation 1

Dino Crisis 2
Final Fantasy VII
Spider-Man
WWF SmackDown 2
Total NBA 97

PlayStation 2

God of War 2
Final Fantasy X
Tony Hawks Pro Skater 4
WWF SmackDown Shut Your Mouth
Darkwatch

PS1:
SmackDown 2
SmackDown 1
WarZone
Attitude

PS2:
SmackDown HCTP
GTA Vice City
SmackDown SYM
SmackDown JBI
Fifa 2003

Ps1
Resident evil

Ps2
GTA
Gran turismo

Looking through here there’s some awesome games.
Hopefully PlayStation get there act together and we have all of these available on download soon.

can't speak much on the PSOne, but here's my list of games that pretty good to start with:

Tony Hawk's Pro Skater 3 and or 4 (THPS 3 is 1st game under the new engine from the PS1 games and THPS4 builds on that, though it's much, MUCH harder, but still fun to play)
Tony Hawk's Underground 1 and or 2 (THUG1 establishes a great story mode with new mechanics that wasn't present in THPS4, and THUG2 is zanier than it's contemporaries, but its fun is expanded upon)
Tony Hawk's American Wasteland (takes the series to a more grounded reality, not in any way realistic, but it's more akin to THUG1's style)
GTA Vice City and or San Andreas (from GTA 3 to the last produced GTA game on the older consoles, which AFAIK it's GTA Vice City Stories, have been stellar to play through)
Bully (another fantastic game by Rockstar; top notch, quality writing, especially for a Teen rated game, not to mention Shawn Lee's involvement as the composer of Bully's amazing soundtrack)
Twisted Metal Black (a more mature and darker approach from the creators of Twisted Metal 1 and 2, with a phenomenal soundtrack to boot as well)

PES seemed to have worked out the perfect formula for that generation of console, the game play was far superior to FIFA and I loved editing so licenses have never been a thing for me.
Playing those games back they feel so sluggish now but for that period of time, they felt perfect. The PS3 era was just awful trying to be more like FIFA and getting it wrong and PS4 era has, for me, been a bit hit and miss.

I can't imagine how many hours I sunk into PES between 2003-2008 on the PS2, probably didn't help my GCSE revision much. The days before just installing an option file that corrects every name and kit and adds new teams in, when you'd find pixelated logos online then recreate them square by square; editing each name 'Von Mistelroum' > Van Nistelrooy; the classic teams - I loved having individual countries' classic teams rather than just a Euro or World. Never got bored of playing cups. My Master League Aston Villa sweeping everyone away and at the end of each season creating a new set of fantasy kits for the next, plus the brilliant little feature of the game adding a star above the badge for each European Cup you won.

I've never had as much fun with any football game since and with the offerings of the past few years, I have doubts I will again.