Korean: How to ask each other's opinion | (으)시겠어요、(ㄹ/을)래요?

A little review of the futuristic expressions learned in Korean: Future Tense 겠, 거예요, 게임요 Usage, which can be used to express "to (do)..." in Korean.
How do you say "Do you want to (do)...?" in Korean?
This article introduces the use of "(으)시겠어요" and "~(ㄹ/을)래요" to ask for the other person's meaning and opinion.
This is a basic and common grammar, so let's learn how to use it.

(으)시겠어요？

When you want to say "Do you want to (do)...? You can use "(으)시겠어요?" when you want to ask the other person's meaning or opinion like this. This is a sentence ending.

When the verb stem ends in a vowel, use "verb stem + 시겠어요?".
When the verb stem ends in a consonant and is not an ㄹ consonant, use "verb stem + 으시겠어요?".
When the verb stem ends in an ㄹ consonant, use "verb stem without the ㄹ ending + 시겠어요?".

(ㄹ/을)래요？

When you want to express "You want to do...? You can use "(ㄹ/을)래요?" when you want to ask the other person's meaning or opinion like this. The endings of such sentences are

When the verb stem ends in a vowel, use "the verb stem + ㄹ래요?".
When the verb stem ends in a consonant and is not an ㄹ consonant, use "the verb stem + 을래요?".
When the verb stem ends in an ㄹ consonant, use "the verb stem with the ㄹ ending + ㄹ래요?".

Sentences

뭐 먹을래요?

what do you want to eat?

뭐 먹으시겠어요?

what do you want to eat?

같이 영화를 볼래요?

Do you want to go to the movies together?

같이 영화를 보시겠어요?

Do you want to go to the movies together?