The star was edged out of contention by Bruno Mars and Kendrick Lamar, who each received multiple nominations.
Jay Z leads with eight nominations, including his first ever appearance in the album of the year list for 4:44, which comes 21 years after his debut.
Sheeran did receive several nominations in the pop categories, however.
Shape of You is up for best pop solo performance, while ÷ is shortlisted for best pop album.
Lady Gaga is also relegated to the pop categories for her recent album, Joanne, while Katy Perry fails to receive any nominations at all.
Overall, the shortlist is very male-dominated, with New Zealand's Lorde the only woman cited for album of the year.
Taylor Swift also missed out on a nomination for her single Look What You Made Me Do, one of the most talked-about records of the year.
Her recent album, Reputation, was released too late to be eligible for an award - but the star does receive recognition for writing Little Big Town's single Better Man, which is up for best country song; and for her duet with One Direction's Zayn Malik, I Don't Want To Live Forever, which featured on the Fifty Shades soundtrack.
Jay Z's haul comes in recognition of his 13th studio album, which finds the star in a vulnerable, reflective mood. The title track confronts his infidelity, while the self-lacerating Kill Jay Z discusses how fame inflated his ego.
However, most critics expect rapper Kendrick Lamar to pick up the coveted album of the year prize. Damn, which highlights his incendiary, virtuoso flow, is his third record to make the shortlist.
Despacito, the hit from Puerto Rican singer Luis Fonsi featuring Justin Bieber and Daddy Yankee, lands three nominations including record and song of the year.
It is the first time a Spanish-language song has been listed in the song of the year category since Los Lobos's cover of La Bamba in 1988.
British artists are hard to find amongst the Grammys' 84 categories. The Rolling Stones' collection of covers, Blue & Lonesome, is cited for best traditional blues album; while Laura Marling's Semper Femina is up for best folk album.
Coldplay, Calvin Harris, Gorillaz, Bonobo and Guernsey-born producer Mura Masa also receive nods, but there is no prospect of a British artist sweeping the board like Adele in 2017 or Sam Smith in 2015.
The next class of Grammy winners will be unveiled on 28 January at Madison Square Garden in New York.
Album of the year
- Childish Gambino - Awaken, My Love!
- Jay Z - 4:44
- Kendrick Lamar - Damn
- Lorde - Melodrama
- Bruno Mars - 24K Magic
- Record of the year
- Childish Gambino - Redbone
- Luis Fonsi - Despacito
- Jay Z - The Story of OJ
- Bruno Mars - 24K Magic
- Kendrick Lamar (pictured below) - Humble
Song of The Year
Luis Fonsi & Daddy Yankee Featuring Justin Bieber - Despacito
Jay Z - 4:44
Julia Michael - Issues
Logic ft Alessia Cara and Khalid - 1-800-273-8255
Bruno Mars - That's What I Like
Best new artist
Lil Uzi Vert
Photo copyright: GETTY IMAGES. Caption: Ed Sheeran has been nominated in the pop categories