Here is a link you should take a look at, it will help you in the future.
Jim's answer is good, but be careful on accepting higher rent. That's a big thing right now that landlords and property managers are getting in trouble for. Many areas have city and/or county laws regarding rent rates and if/when they can be raised.
Here is a prime example of why it is regulated: I have a home I know I can rent for $600. I advertise it for $400 to get people to apply and keep the app fees.
Another example: I know I can rent a home for $600, but advertise it for $550 to entice more people to apply in order to drive the rent up above $600.
Be careful, and if it is legal to accept higher rent I think it's a great idea!
I don't miss this situation at all (PM company handles it now). You can choose whomever you want to rent your house, as long as the decision doesn't violate fair housing. You want to be careful in collecting application fees, and depending on what you have in your contracts/applications as to what the fee is for and if it is refunded can keep you out of or put you into a gray area. I would run one applicant at a time, starting with the person that made me feel like they would be the most reliable in paying the rent. Trying to prove what came in when for applications is next to impossible to prove unless you time stamp everything.
Here is a fun fact: I've had applicants accuse, threaten to sue... all sorts of things based on "discrimination." There is no law against discriminating against an individual. Someone may be of a particular race, but if you don't like them as a person, that's enough to deny their application. This baffles most people but it is true.
Good luck to you, and if you need anything else you can send me a message.