This is a great topic, there is a lot of great feedback on this thread.
First, I have to agree with you that because of sites like Zillow, we are seeing a large amount of response to postings for rentals. This is good & bad, it's good because the exposure is so high, but it's bad because it seems like people are replying to every rental that they see, whether they are really interested or not. For example, I set 7 appointments last night to show an apartment that I have coming available next month. 3 of them never showed up & one called to say that he wasn't able to make it.
For my applications, I use SmartMove. I like their system, it's a very quick turn-around & I like the idea that I don't have to take the application fee from the applicant. I think it makes them feel like the application fee is really going towards their background check & not just some made-up fee that I want to collect from them. Since the fee isn't cheap ($35 per applicant), I normally am up-front with any applicant about what the requirements are ( _ x's Rent, ___ Credit, no Evictions, etc...). I take the applications in the order that they are received, but the tricky part is that, while I have an application outstanding, I generally will not accept another application. I will give the first person the courtesy to fill it out & see if it was approved. I don't want the 2nd person to have to pay their fee if the first person is approved.
This method seems to work with me & I find that my prospective tenants appreciate that I'm up-front & honest with them about the process.
I hope this helps! Best of luck getting a great tenant!