We study a one-phase Bernoulli free boundary problem with weight function admitting a discontinuity along a smooth jump interface. In any dimension
N
≥
2
N\ge 2
, we show the
C
1
,
α
C^{1, \alpha }
regularity of the free boundary outside of a singular set of Hausdorff dimension at most
N
−
3
N-3
. In particular, we prove that the free boundaries are
C
1
,
α
C^{1, \alpha }
regular in dimension
N
=
2
N=2
, while in dimension
N
=
3
N=3
the singular set can contain at most a finite number of points. We use this result to construct singular free boundaries in dimension
N
=
2
N=2
, which are minimizing for one-phase functionals with weight functions in
L
∞
L^\infty
that are arbitrarily close to a positive constant.