Sparse signal models have proven effective in a number of areas such as pattern recognition and wavelet-based image compression. However, due to the nonlinear nature of sparse models, the exact performance of sparse approximation-based methods has been difficult to evaluate. This paper considers an abstract sparse signal model consisting of the union of $J$ known $K$-dimensional subspaces in $\mathbb{R}^N$. For the case when the subspaces are generated randomly, an explicit formula is given for the average approximation error of a Gaussian random vector with respect to the sparse model. In the limit as $N \rightarrow \infty$, the approximation error approaches a simple expression related to the rate-distortion of the Gaussian source. This asymptotic approximation error with random subspaces meets an approximation error lower bound that must hold for any collection of subspaces. The paper also considers the estimation of sparse signals in the presence of additive noise. For the case of random, independent subspaces, simple formulae are derived for the estimation error and probability of subspace detection in terms of the signal dimensions and SNR. In the limit as $N \rightarrow \infty$, it is shown that there is a simple, critical SNR to detect the correct subspace, and that SNR is related to the Shannon capacity of an AWGN channel.