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We present the current status of ongoing searches for molecular hydrogen inhigh-redshift (1.8 < zabs <= 4.2) Damped Lyman-alpha systems (DLAs)capitalising on observations performed with the ESO Very Large Telescope (VLT)Ultraviolet and Visual Echelle Spectrograph (UVES). We identify 77 DLAs/strongsub-DLAs, with log N(HI) >= 20 and z_abs > 1.8, which have data that includeredshifted H2 Lyman and/or Werner-band absorption lines. This sample of HI, H2and metal line measurements, performed in an homogeneous manner, is more thantwice as large as our previous sample (Ledoux et al. 2003) and considers everysystem in which searches for H2 could be completed so far, including allnon-detections. H2 is detected in thirteen of the systems with molecularfractions of values between f=5x10^-7 and f=0.1, where f=2N(H2)/(2N(H2)+N(HI)).Upper limits are measured for the remaining 64 systems with detection limits oftypically log N(H2)=14.3, corresponding to log f<-5. We find that about 35% ofthe 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 H2 is detected-- regardless of the molecular fraction -- in 50% of them. In contrast, onlyabout 4% of the [X/H]<-1.3 DLAs have log f>-4.5. We show that the presence ofH2 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(HI)>=20.8than below this limit (19% and 7% respectively). The overall H2 detection ratein log N(HI)>=20 DLAs is found to be about 16% (10% considering only log f>-4.5detections) after correction for a slight bias towards large N(HI). [truncated]

Molecular hydrogen in high-redshift Damped Lyman-alpha systems: The VLT/UVES database