All MoBos support fan control while some specific support RGB control as well. Though, RGB control MoBos are easy to spot since they are marked accordingly, e.g:
Asus has Aura
MSI has Mystic Light
Gygabyte has RGB Fusion
AsRock has Polychrome
As far as your request goes, you can get 3x PWM RGB fans with your budget. But for software control as well, that doesn't happen with the little money you have. You'll need 72 quid if you want software control as well with your RGB PWM fans and you'll get 2x fans, not 3x,
amazon.uk:
https://www.amazon.co.uk/NZXT-Aer-RGB120-120-Controller/dp/B01LXUN946
You can buy the 3rd fan separately, bringing the total to 95 quid,
amazon.uk:
https://www.amazon.co.uk/dp/B01LZ5EKI2
Without software control, you...