  The dual formulation of planar N = 4 super-Yang-Mills scattering amplitudes
makes manifest that the integrand has only logarithmic singularities and no
poles at infinity. Recently, Arkani-Hamed, Bourjaily, Cachazo and Trnka
conjectured the same singularity properties hold to all loop orders in the
nonplanar sector as well. Here we conjecture that to all loop orders these
constraints give us the key analytic information contained in dual conformal
symmetry. We also conjecture that to all loop orders, while N = 8 supergravity
has poles at infinity, at least at four points it has only logarithmic
singularities at finite locations. We provide nontrivial evidence for these
conjectures. For the three-loop four-point N = 4 super-Yang-Mills amplitude, we
explicitly construct a complete basis of diagram integrands that has only
logarithmic singularities and no poles at infinity. We then express the
complete amplitude in terms of the basis diagrams, with the coefficients
determined by unitarity. We also give examples at three loops showing how to
make the logarithmic singularity properties manifest via dlog forms. We give
additional evidence at four and five loops supporting the nonplanar logarithmic
singularity conjecture. Furthermore, we present a variety of examples
illustrating that these constraints are more restrictive than dual conformal
symmetry. Our investigations show that the singularity structures of planar and
nonplanar amplitudes in N = 4 super-Yang-Mills are strikingly similar. While it
is not clear how to extend either dual conformal symmetry or a dual formulation
to the nonplanar sector, these results suggest that related concepts might
exist and await discovery. Finally, we describe the singularity structure of N
= 8 supergravity at three loops and beyond.
