The RTL8139 is supported for its prevalence, but the chip design has serious problems.

Personally, I go with Tulip-clone chipsets, e.g. the Linksys LNE100TX (which has used a few different chips in its lifetime; current versions are fine, archaic versions had some bugs, but are probably still 'fine' when compared to an 8139). They're all well-supported now, again due to prevalence.

This isn't to say that all Realtek chips are bad- the 8019AS (as found on many generic ISA 10baseT cards) is one of the best NE2000 clones ever made (by virtue of its integrated RAM- if you're using early 3c509s in your 486 router, trade them for generic Realteks and feel the difference!), and the 8029 is a PCI version of same that's been a great boon to alternative platforms trying to provide basic ethernet support (e.g. on the Amiga line, with recently-available PCI busboards). They just shot the food with the 8129/8139, their attempts at modern 100baseT chips.