In Buser's book, Geometry and spectra of compact Riemann surfaces, the term "generic surface" is used several times. For example, he says that
a generic compact Riemann surface is determined up to isometry by its length or eigenvalue spectrum
but the notion is not properly defined in the text. The question is then : what is a generic surface in the mind of Buser ? Is it simply a way to call those surfaces whose geometry is determined by the spectrum ?
As an additional question, in the proof of Wolpert's theorem, where does this "genericity" come into play ? Thank you for your time.
Let me preface this answer that I have not looked at this book.
Let $X$ be a space. A property $P$ is generic on elements of $X$, if the set of all $x\in X$ such that $P(x)$ is true contains an open and dense set. This "quantifies" that "most" of the elements of $x$ satisfy the property $P$.
If I fix a topological surface $S$ (i.e. if it is a torus, sphere...) we can look at the space of Riemannian metrics on $S$. This space has a natural topology and the question if the property "the metric is determined by its spectrum" has a precise mathematical meaning.