I've heard there's some rule about publishers not accepting material that's being considered elsewhere - the first read rule or something. Can anyone name it please?
From what I've seen, publishers usually give a stance on simultaneous submissions. Some are fine with it, some are very strictly against it and some like DM's example are somewhere in the middle. It's worth doing your research before you make a move. Try your first choice for publication first.
If you're talking short stories, you may have written it for a specific anthology submission, in which case your first choice is obvious. Otherwise you'll have your own reason for choosing where to submit first - rates, reputation, acceptance rate etc.
With novels you'll have far better luck getting published if you can secure an agent rather than going it alone. If you can, they will advise you on how and where to submit your work. I'm not sure how the simultaneous submissions thing works with regards to agents, just make sure you read their submission guidelines.
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Another related issue is submitting work which has been published before, either with a publisher or free online on somewhere like Stories Space.
You may see it listed as first publication rights or similar. While it's common, it's not a rule
It is worth keeping stuff you want to publish private until you're ready to submit. By that I mean don't give it away for free; showing it to friends, other writers and test readers is not the same. You may be able to get something published that's already been distributed free, but you may have trouble.
The other thing I've seen is publishers offering different rates depending on whether they are the first to receive it or not. So, if you've given it away free, you might find they won't pay you as much.