First, the word “Sincerely” is considered very formal nowadays. Save this closing word for situations in which you must be extremely formal. An example of this might be when writing to the President of the United States.
The other terms all denote a familiarity of some type. In other words, you have an established connection with the person you are writing to.
“Regards” is a general closing that denotes friendliness and politeness. It can be used in almost any business situation.
“Best regards” might be used when you have a closer connection with the person and wish to augment the level of regards you are sending.
“Kind regards” conveys the deepest connection of the three. It show the reader of your letter that you feel a strong and positive connection to them.
None of the four closing phrases could ever really be considered impolite, but you should use them selectively.
