There are one or two posts back in the thread that are not arguing, and I hope the OP finds them, as I think this thread is now going to carry on and (hopefully) be useful to her.
PaintersR makes a good point, some people seem to want drama in their lives and it might well be that in your Mum's case.
Also some people are just petrified on being alone, so afraid that they will put up with anyone, and practically anything, to avoid it. It could be your Mum's problem.
As well as drama of course there is a kind of teenage rush that one gets when falling in love or first meeting someone - your Mum could be wanting to get this rush with new men, but then I'm afraid, just rubbish at what a real adult relationship is about, especially one involving children. Which means being responsible and mature, and not exposing your kids to men who are not good role models or good fathers.
It may be that her own upbringing had these kind of men in it - 'mad, bad and dangerous to know,' (said of the poet Lord Byron) - do you know if this was this the case?
Sometimes child sexual abuse can skew a person's whole attitude to themselves, resulting in the choice of dangerous or problematic relationships later n life, or problems such as promiscuity.
This current guy she has though, it seems from what you say that he is not a bad person as such, but that he is bi-polar, so to some extent you maybe cannot blame him for how he is, anymore than if he had lost a limb.
And although you can see better choices for her, she is going to pick who she picks anyway, regardless of what her kids think, by the sound of it. Actually to some extent I would reckon she might be better out without ANY man in her life - even a nice one - until she has sorted herself out. But - is she likely to do that? Probably not by the sound of it.
And as I said somewhere way above here, at the end of the day she is your mother, you are the child, not the parent. I understand how you must worry about her, and want her to do better for herself, but you cannot make her. As someone else pointed out, she has to want to do it.
It might be good for you to have insight into why she does some of these things. Just - don't waste your youth and life trying to save her from herself. You won't be likely to succeed, I'm afraid.
You cannot pick your parents, but you can choose to do things better in our own life, and I very much hope that you do.
**hugs** by the way. I am sure that you need them.