I assume you're talking about post-"I'm Your Man" L.C.
I really dig all the phases of his career. I know there are many who "only like his older stuff".
I think there are a bunch of mic/pre combinations. Of course everyone's gonna say 251/1073 type stuff.
What's your budget? Are you gonna rent or buy, or are you booking a studio based on what vocal chain you want?
I think the most important thing to me about recording this kind of vocal is to give the preamp a good amount of gas and don't compress too much. The drama in that kind of vocal performance is about leveraging a small amount of dynamic range to make it sound huge and the range can be sucked right out because it's not like recording Chris Cornell.
I recorded some stuff like that (albeit without Leonard Cohen!) into a Soundelux iFet 7/API 512c and liked the way it turned out. I think I'd prefer a non-tube chain because I wanna hear that dry, fast midrange a bit more.
See if you can try a few different things. But really keep the compression way back, at least at first.
i've also seen him sing into a AKG C12 in a tv-item about him recording his last album on BBC i thought. On some foto's of his first recordings my mother has on an old LP, you can also see him singing in some old RCA LDC's.
I'm pretty sure that he uses Neumann U67 and U87. Preamp is Neve 1272. It was clearly mentioned in one of the articles I bumped across recently. Since I'm kinda fan, it's unmistakenly etched in my cerebral tissue
Hey, there are a couple of mics that I've been trying to get my hands on but I haven't heard them yet. I'm mentioning them because they might be a good place to start.
NPNG + Coles 4038 REALLY gets the Cohen low end at close proximity. It will blow your mind. Perhaps, any pre with a similar impedance would do the same trick, but I've only had success with the NPNG. Perfect match.
Obviously, use a strong pop filter (dual layer recommended) at close range when tracking vocals with a ribbon.