  In this paper, we apply Osgood's criterion from the theory of ordinary
differential equations to detect finite-time singularities in a spatially flat
FLRW universe in the context of a perfect fluid, a perfect fluid with bulk
viscosity, and a Chaplygin and anti-Chaplygin gas. In particular, we applied
Osgood's criterion to demonstrate singularity behaviour for Type 0/big crunch
singularities as well as Type II/sudden singularities. We show that in each
case the choice of initial conditions is important as a certain number of
initial conditions leads to finite-time, Type 0 singularities, while other
precise choices of initial conditions which depend on the cosmological matter
parameters and the cosmological constant can avoid such a finite-time
singularity. Osgood's criterion provides a powerful and yet simple way of
deducing the existence of these singularities, and also interestingly enough,
provides clues of how to eliminate singularities from certain cosmological
models.
