  This work investigates the formation of singularities under the steepest
descent $L^2$-gradient flow of the functional $\mathcal W_{\lambda_1,
\lambda_2}$, the sum of the Willmore energy, $\lambda_1$ times the area, and
$\lambda_2$ times the signed volume of an immersed closed surface without
boundary in $\mathbb R^3$. We show that in the case that $\lambda_1>1$ and
$\lambda_2=0$ any immersion develops singularities in finite time under this
flow. If $\lambda_1 >0$ and $\lambda_2 > 0$, embedded closed surfaces with
energy less than $$8\pi+\min\{(16 \pi \lambda_1^3)/(3\lambda_2^2), 8\pi\}$$ and
positive volume evolve singularities in finite time. If in this case the
initial surface is a topological sphere and the initial energy is less than $8
\pi$, the flow shrinks to a round point in finite time. We furthermore discuss
similar results for the case that $\lambda_2$ is negative.
