  We describe an algorithm to organize Feynman integrals in terms of their
infrared properties. Our approach builds upon the theory of Landau
singularities, which we use to classify all configurations of loop momenta that
can give rise to infrared divergences. We then construct bases of numerators
for arbitrary Feynman integrals, which cancel all singularities and render the
integrals finite. Through the same analysis, one can also classify so-called
evanescent and evanescently finite Feynman integrals. These are integrals whose
vanishing or finiteness relies on properties of dimensional regularization. To
illustrate the use of these integrals, we display how to obtain a simpler form
for the leading-color two-loop four-gluon scattering amplitude through the
choice of a suitable basis of finite integrals. In particular, when all gluon
helicities are equal, we show that with our basis the most complicated
double-box integrals do not contribute to the finite remainder of the
scattering amplitude.
