New classes of descriptor fractional continuous-time and discrete-time linear systems with regular pencils are introduced. Electrical circuits are an example of descriptor fractional continuous-time systems. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and Laplace transformation the solution to the state equation of descriptor fractional linear systems is derived. It is shown that every electrical circuit is a descriptor fractional systems if it contains at least one mesh consisting of branches with only ideal supercondensators and voltage sources, or at least one node with branches containing supercoils. Using the Weierstrass regular pencil decomposition the solution to the state equation of descriptor fractional discrete-time linear systems is derived. A method for decomposition of the descriptor fractional linear systems with regular pencils into dynamic and static parts is proposed. The considerations are illustrated by numerical examples.