You can find fine saws at hobby shops and some hardware stores, but might need to special order for something this fine. (Though I didn't need to) Readers Reid and War Frog mention using Jewelers Saws as another excellent source for finding fine saws. I've also talked about hobby knife sets in the past, though those saws are flimsier than this.
On the following mini, I used the saw to cut at the elbow so that I could reposition the arm. A wider saw would have destroyed a bunch of the arm. This way I don't have to do anything to rebuild it.
Saws are similar. Coarser saws are usually also bigger and wider. The teeth on a saw also tend to curve outwards a bit - larger teeth can get pretty wide.
The other thing is that saw blade are pointed and sharp - but not sharpened like a knife. Saw teeth need to be able to chew across a surface. At the scale of our minis, a saw needs to be fine so that the teeth can catch on it and do their job. Huge teeth won't actually be able to cut much of anything.
T.P.I. (Teeth Per Inch)
Reader Reid mentions "TPI" in his comments and it's worth pointing out. The Teeth Per Inch is a rating of how fine a saw is. You can see in the picture above that there are 3 teeth in 1/8"... that means 3 * 8 = 24 tpi. Wikipedia mentions you can find saw TPIs anywhere from 3 to 32.