I guess it depends per country. In Germany it has become common that the landlord will require a credit score (called Schufa) plus the things you mentioned (3 months of pay slips, references of balances paid from past landlord).
Strong data protection rules have made it that there is now a credit score report which is only giving you a pass/fail so that landlords don't know too many details.
AFAICT Schufa is not like a US credit score. It's just OK/NOT OK, or a list of "infractions".
Which means if you always paid your stuff on time, a virtual +1 is the same as a +1000, and the visible result will just be "OK" and after a while the entries will age out if it was a "NOT OK"
Strong data protection rules have made it that there is now a credit score report which is only giving you a pass/fail so that landlords don't know too many details.