[1][2][3] Singer is noted for his work with Michael Atiyah, proving the Atiyah–Singer index theorem in 1962, which paved the way for new interactions between pure mathematics and theoretical physics.

[4] In early 1980s, while a professor at Berkeley, Singer co-founded the Mathematical Sciences Research Institute (MSRI) with Shiing-Shen Chern and Calvin Moore.

[7] Partnering with British-Lebanese mathematician Michael Atiyah, Singer created a linkage between the fields of analysis, especially differential equations, and topology.

In particular, they resolved a conjecture of Israel Gelfand's on how topological constructs could count the number of solutions of differential equations.

[10] In discussions between mathematician Jim Simons and physicist Yang Chen-Ning in the 1970s, it was found that the Atiyah–Singer theorem has a number of applications to physics.

[4][7] With Richard V. Kadison, he proposed the Kadison–Singer problem in 1959,[11] Inspired by quantum mechanics, it turned out to have reformulations in engineering and computer science.

He had five children: Stephen (born visually impaired), Eliot, and Natasha (with Sheila); Emily, and Annabelle (with Rosemarie).

[7] Singer's brother Sidney was a particle physicist with Los Alamos National Laboratory in New Mexico and predeceased him in 2016.