First of all, thanks a lot to all the people who responsed with links to even more songs that (kinda) sound like something by Bad Religion. Some were more obvious than others, but there's definitely some great stuff in there. Check out the links in the comments below if you haven't done so already.

So here are the answers:

1. Neil Young - Opera Star
This should have been an easy one for all of you: What Can You Do? It's not just the melody of the first 2 lines in each verse, but also note the resemblance if the lyrics in the second verse:
So you stay out all night
Gettin' fucked up in that rock and roll bar

2. Eddie & The Subtitles - American Society
Again this is an easy one and also the most well-known one: 21st Century (Digital Boy). It also shows some resemblance to The Empire Strikes First, though.

3. The Who - I'm Free
It's not the melody and it's not the lyrics; it's a reference made in Along The Way: "Like Tommy you are free (...)". 

4. Credence Clearwater Revival - Who Will Stop the Rain
According to Brett Chasing The Wild Goose is his version of Who Will Stop The Rain.

5. The Beatles - Glass Onion
Again it's not the melody or the lyrics; it's the way they refer to previous songs. BR has done the same thing in 21st Century (Digital Boy): "I tried to tell you about no control (...)" and "and then you told me how bad you had to suffer (...)".

6. Steve Miller Band - Come On Take The Money and Run
According to Greg  Billy Gnosis is Brett's version of this song.

7. Pink Floyd - Hey You
Here it's the re-use of the line "and the worms ate into his brain" in Billy Gnosis:
But it was only fantasy
The wall was too high as you can see
No matter how he tried he could not break free
And the worms ate into his brain

8. The Beatles - You Never Give Me Your Money
Another very clear one (I think). It's the "1, 2, 3, 4, 5, 6, 7, all good children go to heaven" line in Anesthesia.

9. Genesis - The Lamb Lies Down On Broadway
Every BR fan that hears the first line in this song should recognize The World Won't Stop Without You.

10. Badfinger - Without You
It's covered by a lot of other artists, but Mariah Carey's version is probably the most famous. Greg squeezed in the "Can't live if living is without you" line in The Empire Strikes First. Greg sings "Can't live if living is with you", though.

11. King Crimson - 21st Century Schizoid Man
The title of the song should already give it away: 21st Century (Digital Boy).

12. Pink Floyd - Pigs (Three Different Ones)
Not recognizable at first maybe, but if you skip to about 4:10 you'll clearly hear some The Dichotomy in there.

13. The Beatles - We Can Work It Out
The "No time for fussing and fighting my friend" line in You comes from this song by The Beatles.

Hey everyone, Marty here... Since it's BR off-season right now we thought it would be fun to use the front page to spread some little known facts about the band. Let's start with one of my favorites topics: songs by other artists that Greg and Brett "borrowed" melodies, lyrics or ideas from.
We're pretty sure there's more than these, but here's what we could come up with so far. If you know more, please let us know!

So check out the songs below and see if you can recognize the Bad Religion song(s). Remember: sometimes it can just be a single line in the lyrics or a reference. The answers will be posted in a few days.

@Brett, Greg or Jay... if you see this and you think we got something wrong or have more to add, we'd love to hear about it!

1. Neil Young - Opera Star

2. Eddie & The Subtitles - American Society

3. The Who - I'm Free

4. Credence Clearwater Revival - Who Will Stop the Rain

5. The Beatles - Glass Onion

6. Steve Miller Band - Come On Take The Money and Run

7. Pink Floyd - Hey You

8. The Beatles - You Never Give Me Your Money

9. Genesis - The Lamb Lies Down On Broadway

10. Badfinger - Without You

11. King Crimson - 21st Century Schizoid Man

12. Pink Floyd - Pigs (Three Different Ones)

13. The Beatles - We Can Work It Out