As mentioned in another question I am part of a new team that is not performing well.
The main reason is the relatively newly promoted manager (with a bit of "help" from some team members).
The whole thing has gone sour and I personally think I should be at least part of the solution in the following sense.
I have observed what she has been doing wrong and honestly in most cases it is either due to lack of competence, procrastination and lack of leadership skills.
Even if I could figure out a way to give her my feedback on this in a way that a normal person * would find it constructive I am not sure how would I even approach the subject and start giving her unsolicited feedback.
So considering that as an employee I am supposed to support my colleagues what would be the most appropriate way to give her my feedback?
* I use the word normal on purpose and I think it makes sense after reading my other question about her behaviour
I would keep my head down. There's nothing she can or will do about problems you describe. If your description is correct, she may very well be aware of the problems and their causes. If she isn't, you stand even less chance of entering in a constructive discussion. Support your colleagues by doing your job as good as you can, and if you have time to spare, help them. The management problem isn't yours, but your boss's boss.
Do you have regular 1-1 time with your manager? I suspect you don't as it seems so difficult to even get started. So schedule some time for a private chat with her. Then:
Prepare what you want to say in advance.
Don't approach it from the angle of "you're doing it wrong". There may well be factors you're not aware of. Identify one or two specific things to talk about, which affect you, and where you're able to point to a positive, realistic change which would improve your happiness and effectiveness.
Then depersonalise it. Frame it as "I'd like this aspect of team life to get better", not "I'd like you to change".
Don't use it as an opportunity to vent your frustrations, and don't speak for the rest of the team. "We all think..." will put her on the defensive much more quickly.
Try to use it as a starting point for a better relationship with your manager. If nothing else, having a regular time where you can be open and frank about the way the team works is a big step forwards.
Yes, there is a risk this will backfire. But hopefully your manager, being new to the role, will understand that well-intentioned feedback is a chance to improve.
Edit: "So considering that as an employee I am supposed to support my colleagues..." - your manager is a colleague too. Make it clear you are supporting her as she grows into the role - and do support her.