We consider goal-oriented error estimation for free-boundary problems using isogeometric analysis. Goal-oriented methods require the solution of the dual problem, which is a problem for the adjoint of the linearized free-boundary problem. Owing to linearization, this dual problem includes a curvature-dependent boundary condition, which leads to cumbersome implementations if the discrete free boundary is only continuous, as in a piecewise-linear representation. Isogeometric finite elements straightforwardly provide continuously differentiable free boundaries for which the corresponding dual problem can be easily implemented. We illustrate the computation of the linearized-adjoint problems with two test cases and estimate the error in corresponding quantities of interest. In the first problem, a single B-spline patch can be employed. In the second problem, we employ T-splines. Bézier extraction is used to provide a finite element interface to these two distinct spline technologies.