Good Place For Drinks - Beverly/Fairfax Area
Good choices are fairly lacking. All the nice restaurants on Beverly have small bars, if you want to belly up there (Grace, Hatfilelds, Jar). Also Chameu on Fairfax just north of Beverly.
You can also consider the Dime on Fairfax but ONLY if it's before 10 or so. After that the volume soars to unbearable levels (music and crowd).
Also, in the Grove itself, is the Whisper Lounge which is surprisingly nice for drinks, if you don't mind the fact that you're effectively in a mall.
And Celadon on the Southwest corner of Fairfax and 3rd has a nice upscale lounge.