Do something minutely and you'll do it in an attentive or meticulous, careful way. If you minutely edit your English paper, you'll be sure to catch every last mistake.
A detective might investigate a crime minutely, examining every possible clue in an intent way. Minutely can also modify an adjective, as in a minutely detailed news report or a minutely precise dollhouse version of Buckingham Palace. At the root of minutely is the Latin minutus, "little, small, or minute."