Context. C-cyanomethanimine (HNCHCN), existing in the two Z and E isomeric forms, is a key prebiotic molecule, but, so far, only the E isomer has been detected toward the massive star-forming region Sagittarius B2(N) using transitions in the radio wavelength domain. Aims. With the aim of detecting HNCHCN in Sun-like-star forming regions, the laboratory investigation of its rotational spectrum has been extended to the millimeter-/submillimeter-wave (mm-/submm-) spectral window in which several unbiased spectral surveys have been already carried out. Methods. High-resolution laboratory measurements of the rotational spectrum of C-cyanomethanimine were carried out in the 100–420 GHz range using a frequency-modulation absorption spectrometer. We then searched for the C-cyanomethanimine spectral features in the mm-wave range using the high-sensitivity and unbiased spectral surveys obtained with the IRAM 30-m antenna in the ASAI context, the earliest stages of star formation from starless to evolved Class I objects being sampled. Results. For both the Z and E isomers, the spectroscopic work has led to an improved and extended knowledge of the spectroscopic parameters, thus providing accurate predictions of the rotational signatures up to ~700 GHz. So far, no C-cyanomethanimine emission has been detected toward the ASAI targets, and upper limits of the column density of ~10¹¹–10¹² cm^-2 could only be derived. Consequently, the C-cyanomethanimine abundances have to be less than a few 10^-10 for starless and hot-corinos. A less stringent constraint, ≤10^-9, is obtained for shocks sites. Conclusions. The combination of the upper limits of the abundances of C-cyanomethanimine together with accurate laboratory frequencies up to ~700 GHz poses the basis for future higher sensitivity searches around Sun-like-star forming regions. For compact (typically less than 1″) and chemically enriched sources such as hot-corinos, the use of interferometers as NOEMA and ALMA in their extended configurations are clearly needed.