* "fill out" means almost the same as "fill up". "fill out" may intend to convey the meaning of presenting something to somebody else by making it filled, while "fill up" just means the container is full, not necessariely being shown to others.
* "fill" indicates the act; the result is not clear.
"Fill in" is a good synonym of "complete" when we are talking about forms, I think. However, for some unknown reason, I find its use as a participial adjective (filled in) somewhat dicey. Please fill in this form.
The form must be filled in and sent to the National Revenue on a monthly basis.
The filled in form contains lots of errors.
I don't know why I do not want to say that last one even though I suppose it I could easily do so by making it a compound adjective adding a hyphen - the filled-in form. The same goes for "fill out". When I must use a participial adjective, I usually bet on "completed": The completed form must be sent to me.