In this talk, I will present several geometric and analytic characterizations of purely unrectifiable quasicircles. These necessary and sufficient conditions are expressed in terms of various notions such as Dirichlet algebras, harmonic measure, analytic capacity and continuous analytic capacity. As an application, I will explain how to construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion. This answers a question raised by Paul Gauthier.