Context. There is now ample evidence that dust is already present in abundance at high redshift. However, given the faintness of distant galaxies in the optical and the near-infrared, datasets are still limited and how the dust affects the emerging radiation of galaxies at very high redshift is not yet fully understood. Aims. Using the ALMA Large Program to INvestigate [C ii] at Early times (ALPINE), our objective is to quantify the dust attenuation properties in galaxies at z = 4.4-5.5, and in particular the shape of their attenuation curve. Methods. Using the CIGALE Spectral Energy Distribution (SED) code, we model the stellar populations and their interaction with the dust in order to measure some of the physical properties of the ALPINE sample. We select a subsample of 23 main-sequence galaxies requiring a detection in at least 6 bands in the rest-frame UV and optical, as well as in the dust continuum around 158 µm and/or the [C ii] fine structure line, whose inclusion is important to improve the constraints on the physical properties, while having a reasonably small uncertainty on the slope of the attenuation curves. Results. We find that the attenuation curves span a broad range of properties, from curves that are much steeper than the SMC extinction curve, to shallower than the starburst attenuation curve. The shape of the attenuation curves strongly depends on the Vband attenuation. Galaxies with the lowest attenuation also present the steepest curves. The steepness of such curves is probably the consequence of the combination of the intrinsic physical properties of the dust, the relative distribution of stars and dust in the interstellar medium, and the differential reddening, with an important fraction of the dust concentrated in star-forming regions. The broad range of attenuation curves found at z ∼ 5 shows that no single attenuation curve is appropriate for main sequence galaxies and that assuming a fixed curve can lead to large errors, for instance in the interpretation and use of the IRX-β diagram, if SED modeling is not feasible. Conclusions. Great caution should be exercised when correcting high redshift galaxies for the presence of dust using the UV slope β as it can affect the estimation of both star-formation rate and stellar mass even at low V-band attenuation due to the steepness of the attenuation curve. However, when SED modeling can be used, the impact of the choice of the attenuation curve on the star formation rate and the stellar mass is limited.