Besides really long shutter speeds....noise isn't affected by the shutter speed alone.
The two factors that really affect noise are ISO and exposure. Lower ISO will have less noise, that's a no-brainer. Also, a well exposed image will almost always exhibit less noise than a poorly exposed image....especially when you use software (Photoshop) to correct the exposure problem.
Of course, different cameras have different noise characteristics. Larger sensors really help, which is a big reason why DSLR cameras are better than 'digicams', for noise. Technology gets better all the time, which is why the newest DSLR cameras do pretty well at ISO 3200 and are practically noise free at ISO 100.