Beyoncé leads the field, with nine nominations overall, including four for Black Parade, a protest anthem released at the height of the Black Lives Matter protests this summer.
Swift, with six nods, could win album of the year for a record-breaking third time with her lockdown album Folklore.
Dua Lipa also picked up six nominations for the disco-tinged Future Nostalgia.
The British star, who was named best new artist by the Grammys two years ago, was also on hand to read out some of the nominees - including best children's album and best historical album - in an online livestream.
Rapper Roddy Ricch tied with Swift and Lipa on six nominations, the majority of which recognised his hit single The Box, which spent 11 weeks at number one earlier this year.
However, there was disappointment for R&B star The Weeknd, who was completely shut out, despite having the biggest-selling album of 2020 in the US with After Hours.