The original question: 
"Does anyone know of a way of removing/suppressing the stiple-like finish 
found on some photographs? When I try to scan high school yearbook and 
some postcards, the scans result in an image with a distinct pattern on 
the image. As a result, the scan looks much worse than the original. Does 
this require specialized software (Photoshop, etc.)?"

George Willick's answer:

"This is called moire interference and the patterns (variable depending 
on scan dpi) is usually a conflict between screening used by printers 
and your dpi. Moire can also occur on prints made without glossy paper.

High end photo programs have moire interference corrections but like 
most photo programs, the success rate is not altogether high.

I've found that if you xerox a screened image (useful with large 
images), you can then scan that without moire.

Also you can copy a screened image with a digital camera without moire.

The HARD way, is to change scanner dpi until you walk the interference 
into a resonant frequency and the image will scan correctly...usually 
thats a large file, which can be reduced successfully once scanned."

Sherri's addition to the answer:

"Most scanners come with some sort of image/photo manipulating software that
you should be able to use to touch up the scanned images with.

Before that, though, try adjusting the scanning settings. Scan them as a
color photo instead of black and white. This will actually improve the
image quality. Also, I'd suggest scanning at a minimum of 200 or higher
dpi. When saving the images, chose a low compression jpg file (I use less
than 40% if I'm going to want to print, or be really good quality). The
higher the compression, the lower the quality of the image that you'll
actually be viewing when looking at the file or printing it. Each time you
open the image and do anything to it and save it, you're actually decreasing
the quality. 

If I'm working with scanned images making corrections/touch-ups, I always
save the file with a different name so that I have the original image I can
go back to if one of the later versions gets too bad."