The punctuation of the sentence: the dog barked and the bird sang; is correct. these sentence does not need a comma to divide the different ideas due to the fact that it is too short.
coomas tend to be used: before any coordinating conjunction (and, but, for, or, nor, so, yet) that links two independent clauses; after a dependent clause that starts a sentence; to offset appositives from the rest of the sentence.
However, if the clauses are short, then the comma should be left out.