  Consider large signal-plus-noise data matrices of the form $S + \Sigma^{1/2}
X$, where $S$ is a low-rank deterministic signal matrix and the noise
covariance matrix $\Sigma$ can be anisotropic. We establish the asymptotic
joint distribution of its spiked singular values when the dimensionality and
sample size are comparably large and the signals are supercritical under
general assumptions concerning the structure of $(S, \Sigma)$ and the
distribution of the random noise $X$. It turns out that the asymptotic
distributions exhibit nonuniversality in the sense of dependence on the
distributions of the entries of $X$, which contrasts with what has previously
been established for the spiked sample eigenvalues in the context of spiked
population models. Such a result yields the asymptotic distribution of the
sample spiked eigenvalues associated with mixture models. We also explore the
application of these findings in detecting mean heterogeneity of data matrices.
