  We derive an exact solution to the local nonlinear Schr\"odinger equation
(NLSE) using the Darboux transformation method. The new solution describes the
profile and dynamics of a two-soliton molecule. Using an algebraically-decaying
seed solution, we obtain a two-soliton solution with diverging peaks, which we
denote as singular soliton molecule. We find that the new solution has a finite
binding energy. We calculate the force and potential of interaction between the
two solitons, which turn out to be of molecular-type. The robustness of the
bond between the two solitons is also verified. Furthermore, we obtain a new
solution to the nonlocal NLSE using the same method and seed solution. The new
solution in this case corresponds to an elastic collision of a soliton, a
breather soliton on flat background, and a breather soliton on a background
with linear ramp. Finally, we consider an NLSE which is nonlocal in time rather
than space. Although we did not find a Lax pair to this equation, we derive
three exact solutions.
