We consider the problem of reconstructing general solutions to the Helmholtz equation $\Delta u+\lambda^2 u=0$, for some fixed $\lambda>0$, on some domain $\Omega\subset \R^2$ from the data of these functions at scattered points $x_1,\dots,x_n\subset \Omega$. This problem typically arises when sampling acoustic fields with $n$ microphones for the purpose of reconstructing this field over a region of interest $\Omega$ that is contained in a larger domain $D$ (i.e., a room).% in which the acoustic field is defined. In many applied settings, the boundary conditions satisfied by the acoustic field on $\partial D$ are unknown as well as the exact shape of $D$. Our reconstruction method is based on the approximation of a general solution $u$ by linear combinations of Fourier-Bessel functions or plane waves $e_{\bk}(x):=e^{i \bk \cdot x}$ with $|\bk|=\lambda$. We study two different ways of discretizing the infinite dimensional space $V_\lambda$ of solutions to the Helmholtz equation, leading to two different types of finite dimensional approximation subspaces, and we analyze the convergence of the least squares estimates to $u$ in these subspaces based on the samples $(u(x_i))_{i=1,\dots,n}$. Our analysis describes the amount of regularization that is needed to guarantee the convergence of the least squares estimate towards $u$, in terms of a condition that depends on the dimension of the approximation subspace and the sample size $n$. This condition also involves the distribution of the samples and reveals the advantage of using non-uniform distributions that have more points near or on the boundary of $\Omega$. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.