to chase rainbows means to waste your time trying to get or achieve something impossible.
And I found this about "chasing pavements" on the internet:
- it is about not even having anything to chase... so you are just chasing pavements
- to keep pursuing something even if you know that to keep following it, you may not get anywhere at all
- Adele describes ‘Chasing Pavements’ as “It’s me being hopeful for a relationship that’s very much over. The sort of relationship you hate when you’re in it, but miss when you’re not”.
Hope that helps although I am not a native English speaker.