Can you start a sentence with And, Yet, or But?

The coordinating conjunctions (and, yet, but, for, so, or, nor) are supposed to join things. But can they join sentences? That question has been contended for decades. Bryan Garner, however, in the Oxford Dictionary of American Usage and Style, says that the belief that a conjunction cannot start a sentence is a “rank superstition.”  And The Gregg Reference Manual says nothing about any rule against using conjunctions to start sentences, but merely warns against overusing this technique.
So starting a sentence with a conjunction is not a problem. In fact, you can even start a paragraph with one.
The only remaining question is whether to use a comma after the conjunction. The answer is no. But sometimes the conjunction may be followed by a truly parenthetical element surrounded by commas, coincidentally making a comma necessary after the conjunction. If you are putting a comma after the conjunction, make sure the phrase or clause after the comma is truly parenthetical.
Check the comma used in these sentences:

The shoes are comfortable despite the hole in the heel and the scuffed toes. But when it rains, my socks get wet. (“when it rains” cannot be surrounded by commas, because it is essential to the meaning of the sentence: the socks get wet only when it rains.)

The shoes are comfortable and you may still love them. But, as your mom says, they should be replaced. (“As your mom says” is a nonessential independent comment. If you removed it from the sentence, the meaning would not change. The shoes should still be replaced whether your mom says so or not. The two commas indicate that it is nonessential.)

In sentence 2, the comma after the conjunction but is there because of the parenthetical clause. If that clause were not there or were not parenthetical, there would be no comma.
Summary: You can start a sentence with a conjunction, and you should not put a comma after the conjunction. If there is a parenthetical phrase or clause after the conjunction, there might coincidentally be a comma there, but that comma is not due to the conjunction.