Used to ask someone's opinion or suggest something: 'How about...?' or 'What do you think...?'. It has a feminine, softer tone.
このドレス、どうかしら？似合うと思う？
How about this dress? Do you think it suits me?
明日のランチ、イタリアンにしようか。どうかしら。
Shall we do Italian for lunch tomorrow? What do you think?
この色で壁を塗り替えるのはどうかしら？明るくなると思うわ。
How about repainting the wall this color? I think it will brighten things up.
An expression of wondering or uncertainty—'I wonder' or 'I'm not sure'—with a feminine tone.
彼、来るかどうかしら。
I wonder whether he'll come.
この仕事、うまくいくかどうかしらね。
I wonder if this job will go well.
あの店、まだ開いているかどうかしら。
I wonder if that shop is still open.