I'm with SeaSaltMill, it's not a good place.
I am told that 'I think mummy will always hate you', to which I smile and nod, but as a result of how she treats my family she has no idea what hate is compared to my feelings for her. She will die alone, and she will deserve it.
When dealing with her of course I am polite, when talking about her I constantly reinforce positives 'you have nice hair like mummy'. Its funny because I see our child confused a bit about how mum can be so mean about me and me be so nice about mum, but I think that's a good life lesson, and I think she will come to realise along that mum's behaviour isn't very nice or warranted, so much better than if I were to say what I really felt.
The truthfully answer is it doesn't matter how any of us feel about our child's parents, that's their PARENTS. You don't interfere, criticise, or act negatively towards them in front of their child as you will hurt them. For this reason I wouldn't also look to be best friends, this could get a bit uncomfortable if you have a falling out..... If I needed to contact his ex about childcare issues such as a pick up a could do so in a civil way, that's what it should be and I would stick with this. Even in our horrible situation if there's a play or something we all want to go to we can sit together and politely say aw look, she's the baby jesus / whatever.
Keep that politeness and relationship good, but don't get carried away I say, there are other less complicated people to be best friends with but you must respect mum and dad in front of their child.