Aims. We present the current status of ongoing searches for molecular hydrogen in high-redshift (1.8 < z_{\rm abs} \le 4.2) Damped Lyman- \alpha systems (DLAs) capitalising on observations performed with the ESO Very Large Telescope (VLT) Ultraviolet and Visual Echelle Spectrograph (UVES). Methods. We identify 77 DLAs/strong sub-DLAs, with log N (H I) \ge 20 and z_{\rm abs} >1.8, which have data that include redshifted H sub(2) Lyman and/or Werner-band absorption lines. This sample of H I, H sub(2) and metal line measurements, performed in an homogeneous manner, is more than twice as large as our previous sample (Ledoux et al. 2003) considering every system in which searches for H sub(2) could be completed so far, including all non-detections. Results. H sub(2) is detected in thirteen of the systems, which have molecular fractions of values between f \simeq 5\times10 super(- 7) and f \simeq 0.1, where f = 2 N (H sub(2)) /(2 N (H sub(2)) +N(H I)). Upper limits are measured for the remaining 64 systems with detection limits of typically log N (H sub(2)) similar to 14.3, corresponding to log f <-5. We find that about 35% of the DLAs with metallicities relative to solar X/H \ge-1.3 (i.e., 1/20th solar), with X = Zn, S or Si, have molecular fractions log f >-4.5, while H sub(2) is detected-regardless of the molecular fraction-in \sim 50% of them. In contrast, only about 4% of the X/H <-1.3 DLAs have log f >-4.5. We show that the presence of H sub(2) does not strongly depend on the total neutral hydrogen column density, although the probability of finding log f >-4.5 is higher for log N (H I) \ge 20.8 than below this limit (19% and 7% respectively). The overall H sub(2) detection rate in log N (H I) \ge 20 DLAs is found to be about 16% (10% considering only log f >-4.5 detections) after correction for a slight bias towards large N (H I). There is a strong preference for H sub(2)-bearing DLAs to have significant depletion factors, X/Fe > 0.4. In addition, all H sub(2)-bearing DLAs have column densities of iron into dust grains larger than log N({\rm Fe}) _{\rm dust} similar to 14.7, and about 40% of the DLAs above this limit have detected H sub(2) lines with log f >-4.5. This demonstrates the importance of dust in governing the detectability of H sub(2) in DLAs. Our extended sample supports neither the redshift evolution of the detection fraction of H sub(2)-bearing DLAs nor that of the molecular fraction in systems with H sub(2) detections over the redshift range 1.8 < z_{\rm abs} \le 3.