How to cut a cigar?
The first thing you should do is closely examine the "head" of the cigar - this is the closed end that needs to be clipped. Almost all have what is called a "cap" - a bit of tobacco leaf used to close of the end - you should be able to see how far down the length of the cigar the cap goes by inspection. Typically only a 1/4" - 3/8" or so; sometimes much less, and on figurado shapes sometimes quite longer. Anyhow wherever the cap stops is your cutting limit - cut beneath the cap's line or even too close and your cigar will start to unwravel, and as you pointed out this is extremely unpleasant. Typically I cut the minimal possible while trying to open approx. 75%-85% of the cigar end's surface area. Sometimes this means a cut as little as 1/32" down, where other times almost 3/8" - it depends entirely on the individual cigar's roll and cap construction.
The single bladed cheapie cutters that most newbies are given or buy for $3 typically do a very poor job of clipping the cap, and result in crushed, split, and tatterted cuts. One thing to keep in mind when using a guillotine cutter is to line up your cigar at eye level and to them clipped it quickly and decisively - I have found this method to give satisfactory results nearly 100% of the time.
Many smokers swear by the .44 Magnum cutter which is a relatively inexpensive punch that is easy to use, makes a perfect round opening, and completely bypasses the problem of how much to clip. I personally don't use one because they do not work as well on figurado shapes, nor can I get as large an opening as I sometimes would prefer. Crestmark also makes a nice cigar punch that extracts the cut cap from itself. I own both types of punches and think they are excellent products and would recommend that you try one at your tobacconist, it may be ideal for you.
V-Cut clippers are also available, and a few cigar smokers I know think this is the ONLY way to clip your cigar. You don't have to worry about the caps length using this type of cutter since you rest the cigar against it, and it "automatically" takes out a v-notched shaped bit of tobacco of the same size everytime. Personally I hate this type of cut, I find that it tends to build up tar on the edges and that some cigars tend to burn unevenly when cut this way. These cutters typically work better on some sizes than others, depends on the size of the v-notch blade.
Cigar scissors are elegant, but they are difficult to use in my opinion. Plus they are damn near impossible to carry around.
Some people use x-acto blades, swiss army pen knives, their teeth, and so on. How you clip your cigar is a matter of what works best for you. Just keep in mind where the cap ends and you should be fine.