I think that all digital files will show some noise when viewed at 100%, the best way to show us the problem is to convert a RAW to jpg in LR with no NR applied. Then upload a 100% crop from it and a resized copy if the full photo.
To address some of the questions, neither use of a tripod or the lens used should effect the noise levels (though a fast lens and a tripod might help keep the ISO down which would help). Personally I would not worry too much about a bit of visible noise so long as it is not impacting on the overall look of an image - digital noise will show even less when an image is printed. I do not apply any NR unless shooting at ISO1600 or higher, when I do use NR I do it in LR as it seems to work quite well. I agree with not over doing NR, I would rather have a slightly noisy shot with plenty of detail than a smooth shot that ends up looking like a water colour painting. Some cameras have very aggressive in camera noise reduction on jpgs which can look terrible which is why I would always shoot RW do I can control it myself (your 650D shouldn't be bad though).
In order to reduce noise as much a possible in the future always try to get the exposure spot on as recovering an underexposed shot will lead to very noisy results. With my 7D I tend to shoot with +1/3rd EV comp as the metering tends to slightly underexpose. I would prefer to recover highlights as this won't add nose and processing RAWs in LR you should have a couple of stops recover in blown highlights. S long as you still have the RAW files of your older shots you can always reprocess them in LR to get the best out of them.
Anyway put up some examples of the problems and hopefully someone will be able to tell if it is normal or a problem with the camera.
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