Psalm 90 is unique in that it is the only psalm that has a superscription that identifies it as having been written by Moses.
Mark S. Smith says in 'Taking Inspiration', published in Psalms and Practice (edited by Stephen Breck Reid), page 245, that the scholarly consensus is that the superscriptions we see on many of the psalms are prose additions to the prior written poems. He cites Brevard S. Childs, who says:
"The Psalm titles do not appear to reflect independent historical tradition but are the result of an exegetical activity which derived its material from within the text itself."
In other words, later Jewish scribes attributed psalms to the person in biblical legend whom they thought most likely to have written the psalm. Childs suggests that superscriptions are to be dated between the Book of Chronicles, which does not cite the superscriptions in citing the psalms, and the Cave 11 Psalms scroll which contains them. Smith (page 257) says that in the case of Psalm 90, the superscript is sometimes missing.
Bob Deffinbaugh, at bible.org believes that Psalm 90:13, which asks God how long will it be before he repents of punishing his servants, shows that the psalm was written during the forty years of wandering, but this is incongruous because the Book of Numbers says that after only two years, Moses already knew how long (Numbers 14:33: "And your children shall wander in the wilderness forty years..."). Psalm 90:13 better suits the time of the Babylonian Exile, when the Jews really did not know how long their punishment would be.
Another objection to Moses as the author of Psalm 90 is that the Bible says he lived to the age of 120 years and that his father, Amran, lived to 137 years. Surrounded by people with lifespans well over 100 years, Moses could not have written Psalm 90:10, which places a virtual upper limit of 80 years on the human lifespan: "The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow; for it is soon cut off, and we fly away."