  The summatory function of a $q$-regular sequence in the sense of Allouche and
Shallit is analysed asymptotically. The result is a sum of periodic
fluctuations for eigenvalues of absolute value larger than the joint spectral
radius of the matrices of a linear representation of the sequence. The Fourier
coefficients of the fluctuations are expressed in terms of residues of the
corresponding Dirichlet generating function. A known pseudo Tauberian argument
is extended in order to overcome convergence problems in Mellin--Perron
summation.
  Two examples are discussed in more detail: The case of sequences defined as
the sum of outputs written by a transducer when reading a $q$ary expansion of
the input and the number of odd entries in the rows of Pascal's rhombus.
