  The paper introduces an adaptive version of the stabilized Trace Finite
Element Method (TraceFEM) designed to solve low-regularity elliptic problems on
level-set surfaces using a shape-regular bulk mesh in the embedding space. Two
stabilization variants, gradient-jump face and normal-gradient volume, are
considered for continuous trace spaces of the first and second degrees, based
on the polynomial families $Q_1$ and $Q_2$. We propose a practical error
indicator that estimates the `jumps' of finite element solution derivatives
across background mesh faces and it avoids integration of any quantities along
implicitly defined curvilinear edges of the discrete surface elements. For the
$Q_1$ family of piecewise trilinear polynomials on bulk cells, the
solve-estimate-mark-refine strategy, combined with the suggested error
indicator, achieves optimal convergence rates typical of two-dimensional
problems. We also provide a posteriori error estimates, establishing the
reliability of the error indicator for the $Q_1$ and $Q_2$ elements and for two
types of stabilization. In numerical experiments, we assess the reliability and
efficiency of the error indicator. While both stabilizations are found to
deliver comparable performance,the lowest degree finite element space appears
to be the more robust choice for the adaptive TraceFEM framework.
