There's no surviving evidence of rites according to lunar phases for Selene, or Artemis, who later supplanted her. Whether they paid attention to full moon, crescents, or new, we'll never know. Statues of Selene have depicted her with crescents in her hair, but there's no indication whether they are decorative or whether they represented a particular period of worship sacred to her. The Greeks seem to attribute less to the moon's phases as a function of time than as an object of light and brightness.
Hecate, the goddess of magic, witchcraft, and crossroads, may have been worshipped during dark moon phases, but this largely appears to be a modern conception grafted onto her by neopagans. In fact her associations with the moon may entirely be modern.
Selene is yet another interesting character. She was known to have many love affairs, but she was deeply in love with Endymion who was a Shepherd Prince. Selene is depicted many times either as riding a bull, or riding a horse side saddle.