How can we forget Sammy Davis Jr.?
Robert Duvall (?) I just watched Tender Mercies recently so that’s where I get singing. I might not rush to buy his Christmas album, but he did sing capably. I know he does the tango really well. Does that gain him entry into the club.
Kristin Chenoweth starred in Wicked for several years, I imagine she didn’t get the role by standing still. 
Beyonce in Dream Girls…much better acting…
However, I can’t believe you people overlooked:
Dick Van Dyke
Julie Andrews
and…wait for it…
Prince! Wait, he can’t act…sorry about that.
Is it Hollywood’s lack of ability to make films requiring singing and dancing or a dearth of younger actors and actresses that can sing and dance? The people mentioned so far were in their 20s and 30s when they acted, sang and dance.
What about Madonna?
Even as a huge, obsessive, ridiculous Madonna fan, I have to say no. In my opinion, of course. But no. She can’t act. She just can’t. She’s had flashes of film brilliance (Desperately Seeking Susan, A League Of Their Own, Evita) but if we’re looking for people who can “really act”, I don’t think we can include Madonna.
ETA: However I will not hear a word against her singing or dancing abilities! I’m armed with YouTube videos! I’ve got clips from Evita and I’m not afraid to use them!
I’d have to agree with others; Lopez is a perfectly good actress and has acquitted herself well in some fine movies against fine actors. Her skills as an actress have been forgotten in the wreckage of overexposure, much of it self-inflicted, but her resume has some first rate performances. She’s terrific in “Out of Sight,” for instance. But she really isn’t a talented singer.
Conversely, Beyonce absolutely can sing, but cannot act, and you can see in the “Single Ladies” parody already linked to that she’s struggling to even be good by SNL’s miserable standards. Next to real actors like Paul Rudd and Justin Timberlake she looks like a volunteer from the audience.
Beyoncé’s problem is similar to Madonna’s - she such a larger-than-life personality and she’s so famous, that instead of becoming a character, she never gets much further than Beyoncé attempting to become a character. You don’t look at the screen and see a character, you think “Hey it’s Beyoncé!”
Or at least I do anyway.
An example of a larger-than-life personality who was able to overcome this is Cher - even as someone who squees every time they see Cher do anything at all, I was able to believe her as her characters in every film I’ve seen her in, and the Academy Award seems to back me up. As I said earlier, it’s a shame she can’t dance or she’d be perfect for this thread.
Frank Sinatra and Sammy Davis Junior could manage all three. Frank won an academy award for acting.
Mandy Patinkin has a great voice, is a very good actor, and he can dance (and swordfight!)
Kevin Bacon tours in a rock band, was one of the iconic dancers of the 80’s, and can act.
Robert Downey Jr. released an album of jazz songs which he sang and performed on piano, and which was quite well received. I think he danced in Chaplin, didn’t he? He’s a great actor, too.
Gregory Hines
Half the cast of Glee.
Howard Keel, Kathryn Grayson, Robert Preston, Jane Powell, Bing Crosby, Bob Hope…
Ben Vereen, Glen Close and William Daniels
Donald O’Connor blows away Gene Kelly and Debbie Reynolds in Singin’ in the Rain.
Phil Silvers
Although I’d pay to see “Tony Robbin’s Broadway”, he won the Tony for “Jerome Robbin’s Broadway”.
I remember Alexander starring in a television version of Bye Bye Birdie some years ago and being amazed how light on his feet he was. His co-star was Vanessa Williams who also deserves a place in this thread.
Barry Bostwick
Roy Scheider
Mark Indelicato
John Travolta
She won’t be breaking into opera any time soon, but Penélope Cruz is the family’s “bad ballet dancer” and does dance and sing decently; I haven’t heard her sister Mónica sing but she was a ballet dancer for over ten years before someone offered her an acting role.
Sure, he’s dead, but Michael Jackson.
Why has no one mentioned Richard Gere or Rosario Dawson?