Warrants will typically show up in background checks, but they do take a bit of digging to find. Bench warrants are issued by a court to authorize police to make immediate arrests when a person does not appear in court. These will be likely to show up on background checks.
Criminal warrants, however, are issued by law enforcement. Depending on where the warrant was issued and who is checking the criminal records, they may or may not show up in background checks. In some cases, criminal warrant information is only accessible to law enforcement officials.
Finally, civil warrants are issued by a judge and included in court records, which means you can expect them to appear in background checks.