  Motivated by the study of kernels of bilinear pseudodifferential operators
with symbols in a H\"ormander class of critical order, we investigate
boundedness properties of strongly singular Calder\'on--Zygmund operators in
the bilinear setting. For such operators, whose kernels satisfy integral-type
conditions, we establish boundedness properties in the setting of Lebesgue
spaces as well as endpoint mappings involving the space of functions of bounded
mean oscillations and the Hardy space. Assuming pointwise-type conditions on
the kernels, we show that strongly singular bilinear Calder\'on--Zygmund
operators satisfy pointwise estimates in terms of maximal operators, which
imply their boundedness in weighted Lebesgue spaces.
