Very tiny number, oddly, neatly divided by 1001 (7)
I believe the answer is:
nominal
'very tiny' is the definition.
'nominal' can be an answer for 'tiny' (thesaurus). I'm unsure of the 'very' bit.
'number oddly neatly divided by 1001' is the wordplay.
'number' becomes 'no' (abbreviation).
'oddly' indicates alternate letters (letters in the odd-numbered positions are used).
'divided by' indicates putting letters inside (inserted letters divide the word).
'1001' becomes 'mi' (I've seen this in other clues).
The alternating letters of 'neatly' are 'nal'.
'no'+'nal'='nonal'
'nonal' enclosing 'mi' is 'NOMINAL'.
Can you help me to learn more?
(Other definitions for nominal that I've seen before include "Token - lion-man (anag)" , "not real" , "Ostensible" , "trifling" , "In name only or token payment" .)