  A few Galactic classical Cepheids were observed in the programmes of space
missions as Coriolis, MOST and Kepler. An appealing opportunity was to detect
additional nonradial modes, thus opening the possibility to perform
asteroseismic studies and making the pulsational content of Galactic Cepheids
more similar to that of Magellanic Clouds ones. However, only hints of
cycle-to-cycle variations were found, without any strict periodicity. In this
context the potential of the CoRoT exoplanetary data base was not fully
exploited despite the wide area covered on the Galactic plane. Therefore, we
investigated all the candidate Cepheids pointed out by the automatic
classification of the CoRoT curves. At the end we could identify seven
bona-fide Cepheids. The light curves were investigated to remove some
instrumental effects. The frequency analysis was particularly delicate since
these small effects can be enhanced by the large amplitude, resulting in the
presence of significant, but spurious, peaks in the power spectrum. Indeed, the
careful evaluation of a very attracting peak in the spectra of CoRoT
0102618121, allowed us to certify its spurious origin. Once that the
instrumental effects were properly removed, no additional mode was detected. On
the other hand, cycle-to-cycle variations of the Fourier parameters were
observed, but very small and always within 3 sigma. Among the seven Cepheids,
there are two Pop. I first-overtone pulsators, four Pop. I fundamental mode
pulsators, and one Pop. II star. The CoRoT colours allowed us to measure that
times of maximum brightness occur a little earlier (about 0.01 period) at short
wavelengths than at long ones.
