Aims. We present the current status of ongoing searches for molecular hydrogen in high-redshift (1.8 < z abs ≤ 4.2) Damped Lyman-α 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) ≥ 20 and z abs > 1.8, which have data that include redshifted H 2 Lyman and/or Werner-band absorption lines. This sample of H i, H 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 2 could be completed so far, including all non-detections.Results. H 2 is detected in thirteen of the systems, which have molecular fractions of values between f 5 × 10 −7 and f 0.1, where f = 2N(H 2 )/(2N(H 2 ) + N(H i)). Upper limits are measured for the remaining 64 systems with detection limits of typically log N(H 2 ) ∼ 14.3, corresponding to log f < −5. We find that about 35% of the DLAs with metallicities relative to solar [X/H] ≥ −1.3 (i.e., 1/20th solar), with X = Zn, S or Si, have molecular fractions log f > −4.5, while H 2 is detected -regardless of the molecular fraction -in ∼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 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) ≥ 20.8 than below this limit (19% and 7% respectively). The overall H 2 detection rate in log N(H i) ≥ 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 2 -bearing DLAs to have significant depletion factors, [X/Fe] > 0.4. In addition, all H 2 -bearing DLAs have column densities of iron into dust grains larger than log N(Fe) dust ∼ 14.7, and about 40% of the DLAs above this limit have detected H 2 lines with log f > −4.5. This demonstrates the importance of dust in governing the detectability of H 2 in DLAs. Our extended sample supports neither the redshift evolution of the detection fraction of H 2 -bearing DLAs nor that of the molecular fraction in systems with H 2 detections over the redshift range 1.8 < z abs ≤ 3.