  We propose a definition of persistent Stiefel-Whitney classes of vector
bundle filtrations. It relies on seeing vector bundles as subsets of some
Euclidean spaces. The usual \v{C}ech filtration of such a subset can be endowed
with a vector bundle structure, that we call a \v{C}ech bundle filtration. We
show that this construction is stable and consistent. When the dataset is a
finite sample of a line bundle, we implement an effective algorithm to compute
its persistent Stiefel-Whitney classes. In order to use simplicial
approximation techniques in practice, we develop a notion of weak simplicial
approximation. As a theoretical example, we give an in-depth study of the
normal bundle of the circle, which reduces to understanding the persistent
cohomology of the torus knot (1,2). We illustrate our method on several
datasets inspired by image analysis.
