This article aims to provide the equivalent criteria for the distinguishability of linear descriptor systems (LDS). Regularity of the matrix pencil, which, loosely speaking, guarantees the existence, and uniqueness of the solution of LDS for any inhomogeneity, is required in this article. A characterization of observability for LDS in terms of distinguishability is given. The Laplace transform together with the Cayley-Hamilton theorem exploited to derive Hautus-type criteria for the distinguishability. In addition, we present examples of distinguishable systems.