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A new class of singlet superconductors with a gap function ∆(k, ωn) which is odd in both momentum and Matsubara frequency is considered. Some of the physical properties of this superconductivity are discussed and it is argued that: i) the electron-phonon interaction can produce this kind of pairing, ii) in many cases there is no gap in the quasiparticle spectrum, iii) these superconductors will exhibit a Meissner effect. PACS Nos. 74.20-z; 74.65+n; 74.30 Ci. February 2008 Some recent models of high-Tc superconductors with unusual structure of the gap function ∆(k, ωn) have introduced general questions about the possible symmetry types of the gap for singlet superconductors. For example, Mila and Abrahams1 discussed a singlet superconductor with a gap which is an odd function in (k − kF ). This form, as discussed by Anderson,2 annihilates the effect of strong short-range repulsion. A careful symmetry analysis leads us to the conclusion that in addition to the standard BCS-like singlet gap function, there is a new, apparently unnoticed, class of singlet superconductors, whose gap function ∆(k, ωn) and anomalous Green’s function are odd in both Matsubara frequency ωn and momentum k. Nearly two decades ago in a little-noticed article, Berezinskii3 considered the possibility of unusual S = 1 triplet pairing in 3He. He argued that it is permissible, from the point of view of symmetry of the superconducting gap, to have a phase in which the gap function is a vector in spin space for triplet case, ∆(ωn,k), odd in Matsubara frequency and even in momentum k. Although it is now commonly believed that, in the observed phases, the gap in superfluid 3He is even in frequency and odd in k, there is no symmetry restriction which prohibits the phase proposed by Berezinskii. We shall adapt Berezinskii’s approach3 to the singlet case. We introduce the anomalous Green’s function in d-dimensions