How many times did they sing live on the show, as opposed to lip-syncing to a recorded track? I know it was rare. All I can think of offhand is Moonlight Bay from Hustling the Hustler, because I saw it a couple days ago. Or just name a song they did and we can figure it out that way. There's one that I still can't tell about, after seeing it many times.

Sounds like in the "Like a Sister" episode, Vic Damone was singing live. Although Dick Van Dyke and Rose Marie were almost surely faking their piano playing.

How about when Dick & Mary sing "Beans can get no bean-ery, in the beanery...?"

That was lip-syncing, because Mary somewhere mentioned it was sung so fast she never got it perfect and different takes had to be spliced together.

On a recent repeat, The Secret Life Of Buddy and Sally, I was surprised that Rose and Morey were singing everything live in their resort act, and Rose's Come Rain or Come Shine sure sounded live too. But Mary and Dick seemed to be lip-syncing when they did Harmony. Their voices were too out front and didn't fluctuate a bit as they danced.
I don't think Mary was as comfortable singing live on camera as the others, and liked pre-recording. Although maybe it was because they were dancing too.

But the one that I can't decide on was on MeTV last night and has Mary singing and dancing to True, Man, True. The episode with Bob Crane. Did she sing it on camera, without pre-recording it first? Her voice isn't as out front and sounds more live, but then the sound of her voice doesn't waver a bit as she dances. But Bob Crane was a drummer and plays a bongo drum and also gives out small yelps during his playing and they all sound like they were recorded live on camera. So did Mary sing live on camera for this one? I still can't tell. Anyone else? I'd probably guess it was lip-syncing but, if it was, can't figure out how Crane's little yelps seem both live and ad-libbed.