Theta characteristic on curves of genus zero and one

The aim of that question is how to compute theta characteristic (i.e a line $L$ bundle such that $L^{\otimes2}=\omega_C$ where $\omega_C$ is the canonical divisor of my surface) on complex compact riemann surfaces of genus zero, one and two.
If i take $\mathbb{P}^1$, the complex riemann sphere, every point $p$ satisfy the relation $-2p=\omega_C$. So this is the trivial case and should i say that i have no theta characteristic on the Riemann sphere? Than if the genus of my surface is greater than zero i see that i have $2^{(2g)}$ theta characteristic. How can i compute theese when $g=1$ or $g=2$?