On a DSLR, most of the sound people call “shutter noise” is actually the mirror flipping up and down, so there’s only so much you can do. Practical options mentioned were:

• Use a camera’s quiet mode if it has one. This usually doesn’t eliminate noise, but may spread the mirror/shutter actions out so the sound is less intrusive.
• Use Live View. Because the mirror is already up, this can reduce the mirror-slap portion of the noise on some cameras.
• For serious sound reduction, use a camera blimp/sound enclosure. That’s the classic solution, though it can limit control access and needs care for heat buildup.
• Avoid DIY damping inside the camera; changing mirror cushioning can affect operation and alignment.

If quiet shooting is a major requirement, some cameras are noticeably softer than others, and mirrorless cameras are often quieter because they don’t use a flipping mirror. But with a typical DSLR like the D90, some mechanical noise is simply part of the design.