Medium resolution (λ/-λ = 5000−10 000) VLT-ISAAC M-band spectra are presented of 39 young stellar objects in nearby low-mass star forming clouds showing the 4.67 μm stretching vibration mode of solid CO. By taking advantage of the unprecedentedly large sample, high S/N ratio and high spectral resolution, similarities in the ice profiles from source to source are identified. It is found that excellent fits to all the spectra can be obtained using a phenomenological decomposition of the CO stretching vibration profile at 4.67 μm into 3 components, centered on 2143.7 cm−1, 2139.9 cm−1 and 2136.5 cm−1 with fixed widths of 3.0, 3.5 and 10.6 cm−1, respectively. All observed interstellar CO profiles can thus be uniquely described by a model depending on only 3 linear fit parameters, indicating that a maximum of 3 specific molecular environments of solid CO exist under astrophysical conditions. A simple physical model of the CO ice is presented, which shows that the 2139.9 cm−1 component is indistinguishable from pure CO ice. It is concluded, that in the majority of the observed lines of sight, 60−90% of the CO is in a nearly pure form. In the same model the 2143.7 cm−1 component can possibly be explained by the longitudinal optical (LO) component of the vibrational transition in pure crystalline CO ice which appears when the background source is linearly polarised. The model therefore predicts the polarisation fraction at 4.67 μm, which can be confirmed by imaging polarimetry. The 2152 cm−1 feature characteristic of CO on or in an unprocessed water matrix is not detected toward any source and stringent upper limits are given. When this is taken into account, the 2136.5 cm−1 component is not consistent with the available water-rich laboratory mixtures and we suggest that the carrier is not yet fully understood. A shallow absorption band centered between 2165 cm−1 and 2180 cm−1 is detected towards 30 sources. For low-mass stars, this band is correlated with the CO component at 2136.5 cm−1, suggesting the presence of a carrier different from XCN at 2175 cm−1. Furthermore the absorption band from solid 13CO at 2092 cm−1 is detected towards IRS 51 in the ρ Ophiuchi cloud complex and an isotopic ratio of 12CO/13CO = 68 ± 10 is derived. It is shown that all the observed solid 12CO profiles, along with the solid 13CO profile, are consistent with grains with an irregularly shaped CO ice mantle simulated by a Continuous Distribution of Ellipsoids (CDE), but inconsistent with the commonly used models of spherical grains in the Rayleigh limit.