I don't own the 60D so can only make broad comments.
One thing to be aware of is that high ISO performance has significantly improved over the years, so an ISO 1600 shot, for example, on a newer camera should be at least as clean as an ISO on a later camera.
And then, to echo some of the comments above, a whole lot rests on both what you are shooting, the light conditions, and how you expose. In the right conditions where you don't have a lot of dark areas to boost in post processing, good light will help to "cancel" the effects of noise. As a result, we have seen plenty of nice shots done at high ISOs, like the one posted above.
Like Dan mentioned, though, this won't always work due to the fact that some images are meant to be printed much larger than a Web shot and the larger you print, the more visible noise will be. Or, you may find yourself shooting things that require close cropping -- wildlife shooting is an example where there just is never enough reach, and so close cropping is part of the "package" and, as a result, you either need great light, or you just end up muttering "the heck with it" and get the shots.
Note that "noisy" shots can be great for Web and personal use because, well, those of us who do this as a hobby may never need to worry about anything else. Or, you can hopefully build up a portfolio of images that are nice and clean and great and are ready to hang in a gallery at a nice print size or to submit to a competition or to sell to a picky stock agency or to be published in a magazine (editors can be known to be picky)!
For me, I did upgrade from my old 30D to my present cameras (the 5DC and the 1D3) largely because I was doing a lot of wildlife shooting in dim light and too much distance to avoid close cropping and I wanted bodies that do a clean 800 and a good 1600 or, if needed, higher. And yeah, I've had to push to ISO 3200 plenty of times when the light goes low. Those shots tend to be a bit painful when looking at them at a high magnification, and those shots will likely not print large so good, but hey, some of them were shots I wanted, and when compressed for a small personal print or for the Web or personal viewing they look nice, so there!
And, Noise Reduction software has been making advances as well. I've been a hard-core Raw shooter and Lightroom user for quite some time and Lightroom has moved ahead in the quality of its noise reduction (especially with the latest release, LR3) and, as of LR2, added "local adjustment brushes". What this means is that in the Raw processing you can take on things such as selective sharpening and softening which, for a high ISO wildlife shot, can make a huge difference! The details in wildlife tend to "drown out" the noise, so if you can get the hang of using a bit of noise reduction then brusing sharpness on the critter and softness on the background areas which often show more noise, than you can get some impressive results! I haven't gotten around to the thousands of wildlife images from the "old days", but...
Of course apps like Photoshop/Elements/Gimp that provide layers, layer masks, and sophistcated way of using them are the "standard" ways of dealing with such "selective" sharpening and noise reduction, but I've just been a bit hard-headed about sticking with Lightroom!