These are the games I successfully converted and work with little to no issues:
Freedom Fighters (USA)
SONIC Unleashed (USA)
Shadow the Hedgehog (USA)
Manhunt (USA)
Ape Escape 2 (USA)
Klonoa 2 (USA)
War of the Monsters (USA)
Bloody Roar 3 (USA)
SEGA Ages - OutRun (JPN)
SEGA Ages - Golden Axe (JPN)
The Mark of Kri (USA)
Urban Freestyle Soccer (EUR)
The Getaway - Black Monday (USA)
Need for Speed - Underground (USA)
Dragon Ball Z - Tenkaichi 3 (JPN)
Battle Stadium D.O.N. (JPN)
SEGA Rally 2006 (JPN)
Maximo: Ghosts to Glory (USA)
State of Emergency (USA)
Marvel Vs. Capcom 2 (USA)
Gran Turismo - Concept 2002 Tokyo-Geneva (EUR)
Tekken 4 (USA)
Super Dragon Ball Z (EUR)
Gran Turismo 4 (USA)
Tekken Tag Tournament (JPN)
18 Wheeler (JPN)
Teenage Mutant Ninja Turtles 3 (USA)
Red Dead Revolver (USA)
And if I had to choose between being able to play PS2 or PS3 games on PS4? PS2, without a doubt.
I especially love Tekken 4, so nostalgic about it.