The most common and most likely answer to this crossword clue is the 10 letter word DISMISSIVE
I've seen this crossword clue in The Daily Telegraph
D
I
S
M
I
S
S
I
V
E
More clues leading to the result DISMISSIVE
- Friend's final letter — is message contemptuous?
- Is girl in sleazy bar, sneering?
- Girl’s letter giving the brush off?
- Is girl in place of ill repute, sneering?
- Letter supporting Germany is lacking consideration
- Diana’s letter is contemptuous
- Taking no notice of detective before second official letter
- Woman’s letter is inconsiderate
- Taking no notice of little girl’s message
- D is letter pointing to door
Buddy explains!

The straight part
We know that "Taking no notice of" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case taking no notice of = dismissive.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does DISMISSIVE mean?
DISMISSIVE adjective- Showing disregard, indicating rejection, serving to dismiss.