  In this paper, we first address the general fractional integrals and
derivatives with the Sonine kernels that possess the integrable singularities
of power function type at the point zero. Both particular cases and
compositions of these operators are discussed. Then we proceed with a
construction of an operational calculus of the Mikusi\'nski type for the
general fractional derivatives with the Sonine kernels. This operational
calculus is applied for analytical treatment of some initial value problems for
the fractional differential equations with the general fractional derivatives.
The solutions are expressed in form of the convolution series that generalize
the power series for the exponential and the Mittag-Leffler functions.
