As a USC student with limited funds, I'm always trying to figure out how to make the most out of my night when I go out. LA is great because there is so much going on for little to no cash if you know where to look. Echo Park/Silver Lake definitely has the most options, see below for my breakdown.
It's easy to hate on Mondays but Monday nights are actually one of the best of the week to go out because Spaceland, Silver Lake Lounge, and The Echo all offer free shows. Most of the bands who play on Mondays are local, up-and-coming acts and it's a great way to see new music. All 3 venues offer ample free street parking if you don't mind a short walk. Just be sure to read the signs thoroughly! If you're down to have a dance party with sexy gay hipster boys, La Cita in Downtown has Mustache Mondays which is loads of fun for just a $5 cover.
For a little pre-game action, you can't beat the Gold Room Special (shot of tequila and PBR on tap for $4 bucks!) at Gold Room in Echo Park. They even give you free tacos! It's small, raunchy, and always a diverse mix of people. Totally awesome and beats paying $6-10 per drink at the venue.
The Echo Curio in Echo Park is also worth mentioning. It's basically the size of a living room with like a 30 person max capacity. Really intimate, lo-fi vibe going on there. It does get LOUD so earplugs are recommended. They don't serve alcohol but there's a liquor store across the street and it's BYOB.
Amoeba Records in Hollywood usually has a free show at least once a week but they start early (7pm) and tend to get way too crowded. Depending on the hype, I'd recommend getting there early or not going at all. Some of the best free shows I've seen have been across the street at Space 15 Twenty. The court yard area has a small stage where their events take place. I caught Ladyhawke and Girls there a few months back and had a great time. In the past, they've also had free movie screenings like the White Stripes documentary (so good).
I stay up to date with the LA music scene by subscribing to a few different blogs, following venues on Twitter, and liking them on Facebook. It takes a little effort but it trumps bumming around at home all night.