  We prove new sign uncertainty principles which vastly generalize the recent
developments of Bourgain, Clozel & Kahane and Cohn & Gon\c{c}alves, and apply
our results to a variety of spaces and operators. In particular, we establish
new sign uncertainty principles for Fourier and Dini series, the Hilbert
transform, the discrete Fourier and Hankel transforms, spherical harmonics, and
Jacobi polynomials, among others. We present numerical evidence highlighting
the relationship between the discrete and continuous sign uncertainty
principles for the Fourier and Hankel transforms, which in turn are connected
with the sphere packing problem via linear programming. Finally, we explore
some connections between the sign uncertainty principle on the sphere and
spherical designs.
