My means of checking:
* ask for four references, two personal, one work, one second to current landlord (current may give a good reference just to have them out!). If I don’t get the references promptly, the person is obviously not interested
* Google everyone
* Google the tenant(s)
* call everyone, talk a lot. See if stories match. See if my amygdala hates the tenant-to-be
* no background/immigration/credit checks other than my own informal one
* one month’s security deposit, no money accepted or contracts given till the above has happened
