Context. Statistical studies of the spectral signatures of jets and winds in young stars are crucial to characterize outflows and understand their impact on disk and stellar evolution. The young, open cluster NGC 2264 contains hundreds of well-characterized classical T Tauri stars (CTTS), being thus an ideal site for these statistical studies. Its slightly older age than star forming regions studied in previous works, such as Taurus, allows us to investigate outflows in a different phase of CTTS evolution. Aims. We search for correlations between the [OI]λ6300 line, a well-known tracer of jets and winds in young stars, and stellar, disk and accretion properties in NGC 2264, aiming to characterize the outflow phenomena that occur within the circumstellar environment of young stars. Methods. We analyzed FLAMES spectra of 184 stars, detecting the [OI]λ6300 line in 108 CTTSs and two Herbig AeBe stars. We identified the main features of this line: a high-velocity component (HVC), and a broad and narrow low-velocity components (BLVC and NLVC). We calculated luminosities and kinematic properties of these components, then compared them with known stellar and accretion parameters. Results. The luminosity of the [OI]λ6300 line and its components correlate positively with the stellar and accretion luminosity. The HVC is only detected among systems with optically thick inner disks; the BLVC is most common among thick disk systems and rarer among systems with anemic disks and transition disks; and the NLVC is detected among systems with all types of disks, including transition disks. Our BLVCs present blueshifts of up to 50 km s−1 and widths consistent with disk winds originating between ~0.05 and ~0.5 au from the central object, while the NLVCs in our sample have widths compatible with an origin between ~0.5 and ~5 au, in agreement with previous studies in Taurus. A comparison of [OI]λ6300 profiles with CoRoT light curves shows that the HVC is found most often among sources with irregular, aperiodic photometric variability, usually associated with CTTSs accreting in an unstable regime. No stellar properties (Teff, mass, rotation) appear to significantly influence any property of protosellar jets. We find jet velocities on average similar to those found in Taurus. Conclusions. We confirm earlier findings in Taurus which favor an inner MHD disk wind as the origin of the BLVC, while there is no conclusive evidence that the NLVC traces photoevaporative disk winds. The [OI]λ6300 line profile shows signs of evolving as the disk disperses, with the HVC and BLVC disappearing as the inner disk becomes optically thin, in support of the scenario of inside-out gas dissipation in the inner disk.