I was skimming through and seeing a lot of solid advice above.
As someone who is still fairly fresh to rentals, going into my 2nd year and 2nd 4-unit, I'm happy to share something that has worked REALLY well for me, in weeding out a large amount of low-quality renters.
I started out by just posting a listing and doing showings and apps as the calls/emails came in... and like the trouble you're having... you can often get a lot of under-qualified applicants coming though.
Something I stumbled into, and others do some variation of this, it's not reinventing the wheel or anything... but for me, it's helped A LOT.
When I make the post/listing now, I will usually now add somewhere that the space is available on "x date" for showing (and make that date 3-4 etc days out). As you start getting you inquiries, by phone, text, email etc, let them know, in case they missed, that the unit isn't ready for showing yet, it will be available "x date" for showing, and right now you are in the process of collecting applications. Get their e-mail address, so you can send one over.
Then email them an application. I use a basic app from Zillow... It's ready to go, and has the main important info. When I send the app via email, I generally have some basic message with it, usually mentioning the fact that you are gathering applications, the ready-to-show date again, that there is no fee due at this time, but after viewing, before moving forward, there would be a basic background/eviction/credit check, which runs $25. Then I usually end it with feel free to contact me with any questions about the space or the application etc...
Even though you're not charging that $25 right then, it tends to weed out A LOT of under-qualified applicants, just because you're making them aware that the background/eviction/credit check is a part of the process.
Now, I'll admit, not everyone will fill out and return the app... but the one's that do... give them the first viewings, if they meet your criteria... you know they're serious. If you don't get a tenant from the people that turned in apps, or sometimes I'll do this right away, but set their showings starting a day after those who did... I send out a mass email to everyone else that has contacted me, letting them that the space is ready to view, and to contact me, if they are still interested...
The cool thing is, for the most part, the one's who are under-qualified still won't jump in, because you've already mentioned that background/eviction/screening check... But, with the others, that maybe just didn't fill out an app because they were uncomfortable/afraid to put their social security number out there or whatever, you've just created a huge "sense of urgency", because they just saw in your email a list of 15 or however many people also included.
You generally get a 2nd wind of apps, and more people (even if they haven't sent an app) wanting to schedule to view it, who are now also worried about those other 15 people getting to it first...
I've had MUCH better success finding tenants, as I've been going about it this way. It helps to pre-screen the bad eggs as well as creating that sense of urgency among the good eggs.
Like I said, nothing I mentioned above is by any means reinventing the wheel... Just coming up with a process that works for you, and makes life easier, through some combination of everyone above's points.
Weeeellll... I didn't intend to write such a novel... But, it was my first response on a BP forum, and for whatever reason, I guess I was a little excited to jump in, lol!
Good luck!