  We prove a Fredholm criterion for operators in the Banach algebra of singular
integral operators with matrix piecewise continuous coefficients acting on a
variable Lebesgue space with a radial oscillating weight over a logarithmic
Carleson curve. The local spectra of these operators are massive and have a
shape of spiralic horns depending on the value of the variable exponent, the
spirality indices of the curve, and the Matuszewska-Orlicz indices of the
weight at each point. These results extend (partially) the results of A.
B\"ottcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spaces to
the case of variable Lebesgue spaces.
