When Donald O'Connor did sommersaults off the walls during his "Make
'Em Laugh" number in "Singin' in the Rain," was he attached to wires?

Hello redtracer.

No, he did not use wires. O'Connor says so himself in this article:
http://64.233.161.104/search?q=cache:CrhCPfyX9tEJ:www.ctv.ca/servlet/ArticleNews/story/CTVNews/1024897385031_20306585/+%22Donald+O%27Connor%22+%22make+em+laugh%22+%22wires%22&hl=en

"The other standout solo of Singin' in the Rain was O'Connor's Make
'em Laugh routine in which he races through clownish skits aimed at
bellylaughs. The climax comes when he seems to defy gravity.

'I had climbed up the wall in two other pictures, but I made only one
revolution,' O'Connor said from his home in Sedona, Ariz. "It was very
difficult to find a finish for Make 'em Laugh. So I decided that I
would climb up a wall and do a back somersault, then I'd climb up
another wall and do a back somersault. Gene said, "Can you do it?" I
said, "I dunno, I've never tried it before."

'So I did it - no wires or anything - and it was wonderful. Two days
later, I went to the set, and everybody was so happy with that number.
And Gene says, "Do you think you could do it again?" I said, "Oh,
sure, anytime." "We're going to have to do it tomorrow," Gene said.
'The cinematographer left the aperture open, and it's all foggy.' So I
did it again.'
---

Hope this is what you are looking for! 

If you need more information, please feel free to ask for a clarification.

I heard that he performed on a concrete floor and that his efforts
left him rather bruised.

It's great to watch though.

Arguably the best scene in the entire film.

But then there were so many memorable scenes.

And that Jean Hagen ... What a gal ...

Did you know that she had to dub for Debbie Reynolds when Debbie was
supposed to be dubbing for her?