To move forward slowly and carefully, typically in order to avoid detection or to investigate something. The detective nosed around the crime scene, looking for any overlooked evidence.
To push or move something with one's nose. The dog nosed the ball towards its owner, eager to play.
To advance or progress by a small margin. The runner nosed ahead in the final stretch to win the race.
To sniff or smell at something. The horse nosed the feed before eating.