That’s a question best left up to your legal counsel. There is much variation between various state and industry laws. However, your industry and job requirements will likely give you a good starting point.
For instance, requiring a motor vehicle records check and/or drug testing is likely appropriate for any employee in charge of operating machinery or driving a company vehicle. If an employee is handling cash or managing cash assets, a credit history check may be appropriate. Volunteer organizations should consider sex offender registry searches. And so on.
No matter what, your first call should be to your legal advisor for input on which types of background checks are appropriate for your organization.