A few people have mentioned submitting to publishers and I know a few like Snowbooks welcome this but how common is this practice in general.

Also, what kind of problems would it create if you were rejected by a publisher but then taken on by an agent, would you need to let them know you'd submitted to and been rejected by the publishers?

Yes, absolutely let them know, or they'll be embarrassed if they try to submit there after.
Not that I have an agent, but I would think your relationship with them should be like your relationship with a business partner: you must be absolutely honest or it won't work. If you're concealing debts (in a metaphorical sense!) you just can't work together efficiently, you have to be able to trust each other.

Yes, you must let an agent know who's already seen it, I would say not at the partial submission stage, but certainly if they ask to see the full MS: they're always reading thinking, 'Now, who would buy this?', and fewer places there are left to send it, the more likely the agent is to feel there's not enough scope left for doing a deal. Yes, if you have to admit that someone's turned it down it does discourage the agent, but your relationship with your agent must to be based on trust and honesty; you can't work together if they don't have the full picture.

Having said that, if you've been rejected directly by named editors, and the interested agent thinks they were all the wrong editors, and there's scope for her to send to the right ones, it could still be okay, sort of. And it's true that if you're made an offer by a publisher, that's a very, very good moment to approach an agent (as long as you haven't gone any distance with the publisher, because as Snowbooks Emma says, it really isn't cricket to do that, and then have the agent waltz in and change everything, or even insist on pulling out).

One pattern that seems to be quite common is to wait until you don't seem to be getting anywhere with agents anyway (but as WWers know, that could/should be 20 or 30, for adult fic) and then try publishers direct, especially the smaller ones who're happy to accept unagented submissions. Some of the big boys (should it be big girls, in such a female industry?) still do as well, of course.

Emma