Ha! That's a great thread idea! There's a bunch of nice friendships on both shows, but my three favorites(and don't be surprised if Xander/Willow and Angel/Cordy aren't there, since it had glaring romantic sides at times)would probably be...
Xander and Buffy
Lots of really nice moments between those two. Obviously the scene in Seeing Red and the follow-ups of the shooting come to mind, but also Xander doing something he didn't want to when Buffy asked him to KO Dawn and get her out of Sunnydale, Xander giving his speech to Buffy in 'Into The Woods', Buffy standing up for Xander in Checkpoint and Entropy ect... But they also argued and almost came to fists at times(see Dead Man's Party and Revelations), which any really good friends do at time. Because you're not afraid to be honest with them.
Lorne and Angel
I'm surprised no one talked about them! I really liked the whole relationship between those two. It started slowly, but once Angel kept coming to see Lorne, they got closer. Lorne was not really you're usual hero haha, and the fact he actually put his life in the balance countless times just to help Angel, that proves a lot to me. Lorne and Cordelia were the two characters that seemed to really care about what was going on inside Angel during the show. I think he actually helped him couple of times to get out of a bad stretch, like in season 2 with the whole firing his friends. Sadly I think Angel hurt their friendship really bad when he asked him to kill Lindsey at the end of s5... He knew Lorne hated violence, so killing a human like that?
Spike & Dawn
Now I have to agree with some peoples who said this wasn't explored enough. I really liked these two together in s5 and s6. The whole thing with Dawn going to see him since she felt 'safe' with him and that he treated her like a real person, I can totally understand. And then Dawn was also treating him pretty well, which I think Spike respected from her. Actually once Buffy got over the whole 'dead' thing, Spike had been a close pal to all the Summers women haha =P Anyway it was nice to see Spike telling Dawn he wouldn't let anything happen to her in Bargaining and the whole scene in the house! Like someone said, felt like he was acting as her big brother.
Now I know there is many more friendships that everyone liked in the Verse, but I picked three that just struck a cord with me =P