Organic isocyanides have an interesting astrochemistry and some of these molecules have been detected in the interstellar medium (ISM). However, rotational spectral data for this class of compounds are still scarce. We provide laboratory spectra of the four-carbon allyl isocyanide covering the full microwave region, thus allowing a potential astrophysical identification in the ISM. We assigned the rotational spectrum of the two cis (synperiplanar) and gauche (anticlinal) conformations of allyl isocyanide in the centimeter-wave region (4-18 GHz), resolved its 14N nuclear quadrupole coupling (NQC) hyperfine structure, and extended the measurements into the millimeter and submillimeter-wave (150-900 GHz) ranges for the title compound. Rotational constants for all the monosubstituted 13C and 15N isotopologues are additionally provided. Laboratory observations are supplemented with initial radioastronomical observations. Following analysis of an extensive dataset (>11000 rotational transitions), accurate ground-state molecular parameters are reported for the cis and gauche conformations of the molecule, including rotational constants, NQC parameters, and centrifugal distortion terms up to octic contributions. Molecular parameters have also been obtained for the two first excited states of the cis conformation, with a dataset of more than 3300 lines. The isotopic data allowed determining substitution and effective structures for the title compound. We did not detect allyl isocyanide either in the IRAM 30 m line survey of Orion KL or in the PRIMOS survey toward SgrB2. Nevertheless, we provided an upper limit to its column density in Orion KL.