Yes, I'm going to assume we're talking about a 60D (not a 6D, which doesn't have a built-in flash).
EDIT: Ah! I see that it is a 60D.
If so, this isn't an uncommon problem with many different Canon DSLR models and there are two or three possibilities:
Usually it's the micro switch in the hot shoe that's causing problems. What's happening is the switch isn't fully returning to it's open position properly, so the camera "thinks" that there is a flash or other accessory mounted in the hot shoe, so is preventing the built-in flash from popping open and hitting a possible obstruction. Usually the camera will give you a warning on the LCD screen when this is occurring.
Fixing it typically couldn't be simpler. Holding the camera normally, look at the hot shoe closely with a flash light, under the leaf spring on the left you should see a tiny plastic "knob" sticking up. What usually happens is the hot shoe spring isn't allowing that pin to fully pop back up into the "open" position.
Just take a small screw driver or wooden toothpick, slide it under the leaf spring toward the rear of the camera to
slightly bend the leaf spring upward a little more.
Note the emphasis on "slight"... it usually doesn't take much and fixes the problem immediately by allowing the little pin of the switch (which is also spring loaded) to more fully pop back up.
Alternatively, it could be that the micro switch is gunked up and sticking or that the little plastic piece has somehow gotten damaged. This website shows a DIY fix for that:
https://shimworld.wordpress.com/2009/02/26/canon-hotshoe-fix/ This usually isn't necessary, though. And more often than not the only reason you'd have to make up a replacement pin would be if you weren't careful during disassembly and lost the original part! However, the above site does have some good illustrations showing what to look for and how the parts fit together (not that you have to do any disassembly, if the above fix works). If you do need to disassemble, you would need precision screwdrivers... And a lot of care not to lose any of the tiny parts. It also might be a good idea to use Locktite on the screws during re-assembly (only a little and the blue type that allows for easy disassembly, if ever needed in the future).
But there also may be other factors. Here's a web site that shows a couple points that may need lubrication (after opening the flash by hand)...
http://www.tutorialgeek.net/2013/02/how-to-fix-popup-flash-on-your-camera.htmlHowever, I don't agree with using WD40 for that purpose. It's actually a pretty poor lubricant and dries out rapidly. Most camera repair techs won't let WD40 anywhere near a camera! There are much better lightweight oils for the purpose (sewing machine oil, for example). I also certainly wouldn't
spray anything in there... That's
very risky, that you'll get the lubricant into places you really don't want it! If nothing else is available but a spray, instead just squirt a small amount of it into a bottle cap or something like that, then use a toothpick dipped in hat to pick up a small droplet and apply that to the points shown in at the above web site.
But, I bet 9 out of 10 times it's the first problem.
And, frankly, a 7D I've used for over five years has done this from new and I've never bothered to fix it. The reason is that built-in flashes are pretty much crap... so I don't use them. The only time I've ever noticed the problem on that camera was when I accidentally bumped the release button and the camera tried to open the flash... not that I was actually trying to use it. Can't recall the last time I used a built-in flash on any camera I own, for that matter.
The problems with built-in flashes are many. First, they're very under-powered... don't give you much reach at all. Also, they rely upon and rapidly drain the camera's main battery... shortening your shooting time a lot. And they're pretty slow recycling. Plus a built-in flash is in just about the worst possible place for redeye and ugly shadow effects!
Even the smallest dedicated accessory flash works much better. It will have more power, has it's own power supply and most will recycle faster. A flash that stands taller in the hot shoe will reduce chance of redeye and improve on shadows to some extent. But for even better results it can be put on a flash bracket and connected to the camera with an off-camera shoe cord, which will further reduce chance of redeye and ugly shadows by moving the flash farther from the lens axis.
Yes, a built-in flash might be handy in an emergency. And many of the more recent cameras' built-in flashes can control an off-camera flash wirelessly or provide focus assist in low light. However, they use a rather obnoxious rapid series of white light flashes to do so... While other accessories such as the ST-E2 flash controller can do the same with a much less noticeable near-IR light communication... Or the newer "R" type flashes and accessories that communicate with each other via radio waves that are both invisible and can work "around corners" as well as over greater distance.
In other words, I highly recommend getting an accessory flash, and just forget about using the built-in! IMO, that's the best solution of all.