Here is my method I got that has done me wonders, I used the method from Brandon Turners book on managing properties:
1.) Full credit/background check: Look for missed payments, and any theft is always a no go.
2.) Call 3 references: Verify that they work there, or that it is an actual person by asking where they work first, and don't mention who you are until after. You're looking for hesitation on how they speak of them as a person they did work/business with depending on a job or rental reference.
3.) Always meet in person: This will give you a good idea of the person, if they're smoking a cigarette while talking to you, or being very negative about their previous landlord make sure you make note of that.
4.) Verify income: Always make sure they make 3 times the money required monthly for rent. I made the mistake of letting someone get in one of my first units that was a bit off the 3 times, they were great tenants very kind and enjoyed their stay and experience with me, but could not afford it when I increased the rent on them, which is something you will have to do because of rising insurance/tax costs.
5.) Let everyone apply, always: Make sure you follow state regulations, and let everyone apply. You risk the chance of being sued for not letting someone apply that asked.
Hope this helps, feel free to reach out if you have any other questions