This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem $$ -left({u'}/{sqrt{1+{u'}^2}} ight)' = lambda a(x) f(u) ; ; ext{in } (0,1), quad u'(0)=0,;u'(1)=0, $$ where $lambdain RR$ is a parameter, $ain L^infty(0,1)$ changes sign, and $f in C^1(RR)$ is positive in $(0,+infty)$. The attention is focused on the case $f(0)=0$ and $f'(0)=1$, where we can prove, likely for the first time in the literature, a bifurcation result for this problem in the space of bounded variation functions. Namely, the existence of global connected components of the set of the positive solutions, emanating from the line of the trivial solutions at the two principal eigenvalues of the linearized problem around $0$, is established. The solutions in these components are regular, as long as they are small, while they may develop jump singularities at the nodes of the weight function $a$, as they become larger, thus showing the possible coexistence along the same component of regular and singular solutions.
