Different phrases suggest different levels of formality (less polite or more polite)

General Invitation Phrases

Fancy (noun)? (very informal)
Fancy …ing? (very informal)

Do you fancy (noun)? (very informal)
Do you fancy …ing? (very informal)

Do you want (noun)? (informal)
Do you want (to…)? (informal)

Would you like (noun)? (polite)
Would you like (to…)? (polite)

How would you like (noun)? (polite, but also suggesting)
How would you like (to…)? (polite, but also suggesting)

Would you be interested in (noun)? (very polite)
Would you be interested in (…ing)? (very polite)

Written Invitations

Whilst spoken English becomes more polite by asking questions, polite written English generally tries to avoid questions.

Instead of giving a question, it usually makes a request or an official invitation.

– examples

Dave and Polly request the pleasure of your company at their mid-summer barbecue on June 21st, starting at 3pm.

Please join us to wish Philip Jones a happy retirement at The Savoy Hotel on Friday August 8th.

You are cordially invited to the wedding of Dave and Polly, taking place on March 16th, at Westminister Abbey.

Exercises

1. Choose phrases to invite these people to dinner:
(i) a friend
(ii) your boss
(iii) a date

2. Write invitations for:
(i) a house-warming party
(ii) a baby shower
(iii) a birthday party
(iv) a stag do / hen night