The phenomenon of the singing cowboy has largely ended. There are a few left, but they don’t seem to be nearly as famous as their predecessors. I couldn’t name any current individuals and only two groups, without looking it up–and one group is a local one, which is the only reason why I know about it.
It might be worthwhile to have 2 categories: individual singers, and groups.
Individuals:
Roy Rogers
Dale Evans
Gene Autry
Groups:
Sons of the Pioneers
That isn’t all of 'em?
John Wayne, just because he was so bad at it.
Riders in the Sky put on a fun concert, if you ever get the chance to attend.
Probably unknown to anyone who didn’t grow up in the DC area in the 50s and early 60s. He was on an episode of the Rifleman though.
I’ll add At The Rainbow’s End as an example from Rex Allen
See, we don’t need polls to figure out how old the board skews.
Or maybe this board is populated by people who like the classics. 
These days, the genre is typically referred to as “Western music”. You will find a long list of Western music artists at that link… one good example I’m familiar with is the Sons of the San Joaquin.
My favorite was Slim Whitman.
People still occasionally use the phrase “Country and Western,” but the “Western” part of country music has been pretty much dead for 50 years.
In my mind, the last successful “singing cowboys” were Marty Robbins and Ian Tyson. Tyson is about the only singing cowboy still alive.
Lets not forget Vicente Fernandez.
Solo: Roy Rogers
Duo: and Dale Evans
Group: Sons Of The Pioneers (early lineup including RR and Pat Brady
or
Riders In The Sky
Don’t like any of them. I can’t even stand the backup singers on “The Life and Legend of Wyatt Earp.” I heard Clint Eastwood sing on “Rawhide” a couple of times. He wasn’t bad.
Would Bob Wills & the Texas Playboys qualify?
I love Tex Ritter. Not only was he a wonderful singing cowboy, but he sired John Ritter, one of my favorite actors.
We tend to forget that other countries have had ranches, cattle, and even singing cowboys.
Anyone outside Australia remember Slim Dusty?
That was after he was put out to stud.
Yep. He was just as good as Roy and Gene.
Autry was my favorite.