All digital images can show objectionable noise, whatever the ISO, if the scene has a higher dynamice range than the sensor. Usually, raising the black point gets rid of it. Black shadows can be nice. But sometimes you might want that detail in the shadows, in which case you need to take 2 different exposures, one for the shadows and one for the highlights, and blend them in PS.
Raising ISO will always result in more noise in the shadows and lower mid-tones, for the same scene. In other words, the signal-to-noise ratio is worse, the higher the ISO. For this reason, there seems little point in using a higher ISO than necessary, unless you want to deliberately create a noisy image for some artistic effect.
The usual reasons for using a higher ISO setting is that it permits use of a higher shutter speed at the same aperture (useful if you are using a long telephoto lens or photographing a quick moving subject which you want to freeze), or it permits use of a smaller aperture at the same shutter speed when increased depth of field is needed.
Sometimes there is a conflict between less noise on the one hand and a potentially sharper image on the other hand. When I got my first DSLR (the D60) I used to err on the side of less noise, until I realised this was crazy. I now prefer sharper images
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