The most obvious ones are things like being a paedophile, a drunk, a drug abuser, being listed on the sexual offenders register, being convicted of a serious crime (e.g. rape, murder, armed robbery). Or of being found to have “unspent” convictions for lesser crimes like theft, fraud, minor assault.
This is by no means an exhaustive list. Having another member of your household who fits anything I described above would also qualify.