Good credit (over 620 credit score), 2 years history of the same job or at least the same type of work and no pets. That opens up just about any rental for you.
Any one of the above not acceptable, then you only have about 20 percent of landlords that will accept not such good credit (if the place is not that great) and willing to take a pet.
If you are a landlord, which judging by your question, you may be, then having a clean apartment, preferably with hardwood floors, central air, laundry (or hookups), basement storage and parking will get you the best tenants. You can advertise on most real estate sites to attract rentals (except Trulia, realtor.com, gsmls.com--you will need a realtor to do that. Since tenants most often pay the realtor fee, it is free for landlords to sign up their place on the multiple listing service with a realtor. Gets you tons more exposure and most qualified tenants are looking with realtors. We do all of the background credit checks and income verification. It's well worth it.