  We propose a three-step program for the classification of stable rank 2
bundles on the projective space $\mathbb{P}^3$ inspired by an article by
Hartshorne and Rao. While this classification program has been successfully
completed for stable rank 2 bundles with even determinant and $c_2\le5$, much
less is known for bundles with odd determinant. After revising the known facts
about these objects, we list all possible spectra and minimal monads for stable
rank 2 bundles with odd determinant and $c_2\le8$. We provide a full
classification of all bundles with positive minimal monads, provide a negative
answer to a question raised by Hartshorne and Rao, and describe new irreducible
components of the moduli spaces of stable rank 2 bundles with odd determinant
and $c_2=6,8$.
