Wands rarely malfunction.
In fact, the only times we have seen in the books where something goes wrong, it was the fault of the wizard or the lack of synergy between wand and the user. Or the wand was physically broken. This seems to be the closest we get to malfunctioning of wands, and here's what happens to them:
- Sir Cadogan's wand broke while he was fighting. Pottermore says this about the incident:
The creature lowered its fearsome head to swallow Sir Cadogan and the pony whole, but the splintered and misfiring wand pierced its tongue, igniting the gassy fumes rising from its stomach and causing the wyvern to explode.
This does prove that wands are still good for something, but also that when they break they misfire.
- Hagrid's wand was broken, though Dumbledore may have mended it. He seemed to be able to use it fine, though Hagrid wasn't the best at magic, since he was expelled in his third year.
- Several wands, upon breaking, were rendered as useless by the user, including Lucius Malfoy's, Neville's first, and Umbridge's. Also, Harry's wand, which was cleanly snapped. Ollivander says of Harry's:
“Yes,” said Harry. “Can you — ?”
“No,” whispered Ollivander. “I am sorry, very sorry, but a wand that has suffered this degree of damage cannot be repaired by any
means that I know of.”
- Ron's wand, of course, breaks. He,interestingly, does not think it useless--probably because he doesn't want to ask his mother for a new one. And when he uses it, it works--it doesn't do what Ron wants, but it does something magical.
I think, then, that we can infer that what happens depends on how big the break is. Cadogan's was only splintered, and it misfired; Harry's broke in half, and did nothing.