(I hesitate to post that question here, but I received on answer on FB:) Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks currently on, look like?
The survey of Nekovar tells you what was known about the Beilinson conjectures in the early 90s. Other surveys/introductions from that time include Scholl-Deninger, Soulé, Ramakrishnan (in Contemporary mathematics 83), and the volume edited by Rapoport, Schappacher and Schneider (introduction here, all articles here). Since then, not a lot has happened I think. There is a fairly recent article of Otsubo with some results for Fermat curves (published version, arXiv version). There is a nice reformulation of the conjectures in terms of Arakelov motivic cohomology, by Jakob Scholbach (see articles on his webpage). You might also want to check out some of the articles of Rob de Jeu and his coauthors. In addition, there have been various attempts at new descriptions of the Beilinson regulator, most recently by Bunke and Tamme (their work will by the way be the topic of a summer school in Freiburg in July), but this does not in itself imply any progress on the Beilinson conjectures themselves.
If you are interested in recent progress on special values in general, there are other areas where more exciting things are happening. Search for work on the Birch and Swinnerton-Dyer conjecture, the Equivariant Tamagawa number conjecture, and Weil-etale cohomology (starting point for Weil-etale: webpage of Baptiste Morin).
In addition to Andreas's excellent answer, we should also mention the Tamagawa number conjecture of Bloch and Kato, which predicts the undetermined rational factor arising in Beilison's conjectural description of the $L$-value. The Bloch-Kato conjecture was later reformulated and generalized by Fontaine and Perrin-Riou to the case of motives with coefficients in an arbitrary number field. Here are some references :
Bloch, Kato, L-functions and Tamagawa numbers of motives.
Fontaine, Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L.
Colmez, Fonctions L p-adiques.