According to Griffith's Quantum Mechanics, "$E$ must exceed the minimum value of $V(x)$, for every normalizable solution to the time independent Schroedinger equation"
As an example, there is no acceptable solution to the time-independent Schroedinger equation for the infinite square well with $E = 0$ or $E < 0$.
For the Dirac delta function potential, however, there are both bound states with $E < 0$ and scattering states with $E > 0$ allowed. This is also true for the finite square well.
Where in the process of solving the T.I.S.E. can one determine if states with $E < V_0$ are allowed?