I took my wife, Valerie, to the emergency room once. She was having severe abdominal pain. It turned out to be from a kidney stone, but she and I were kept waiting nearly four hours before the preliminary diagnostics were finally done, she was given a shot for the discomfort, and a doctor at last saw her and told her what was going on.
Calling businesses or professional offices provides yet more opportunities for waiting. As a rule, these can last from only a minute or so to around a half-hour, after which one may or may not be cut off and need to start all over again.
When going out to eat, one may wait a number of minutes to be seated, yet more time for one's drink order to be taken, then additional time passes pending one's food order being noted. One waits to get the food, next to receive one's check, and after that to have the credit card returned.
Travel affords multiple new chances for long or short waits, starting with the first traffic jam on the way, the line for check-in, the security check, the wait for boarding, another for actually getting underway, etc. Once at a destination, there are often fresh waits for a rental vehicle, accommodations, tickets, entries to parks, museums, plays, or concerts, and so forth.
And these are all but routine waits. Unanticipated waiting can occur at any time. I have heard of planes having unscheduled delays of several hours, Amtrak waits of over a day, delays in getting one's luggage of up to a week, and power outage delays or the computers all being down indefinitely. While driving in South Dakota, my trip west was put on hold for more than 24 hours due to forest and grass fires near or across the interstate. Personal auto failures, such as from a dead battery or an inability to shift out of "park," can occur completely without warning and usually must be followed by waiting for assistance or towing, unplanned repairs, and so on.
Overall, though ordinarily in drips and drabs, our cumulative waiting likely averages a week or more a year, enough for an extra vacation if we could but see it that way.
Women of traditional backgrounds or values may have contended with still greater amounts of waiting, on a guy calling her for a date, for instance, or later for a proposal from Mr. Right.
Let us not forget as well the existential waits, as in "Waiting for Godot." In a sense, then, all our days are spent in waiting.
So, as I was delayed of late pending my new physical therapist taking a look at a sore shoulder (a possible rotator cuff tear, per my doctor, who had that time only required me to linger for 20 minutes), I had pondered ways to fruitfully occupy my time, while waiting.
Different strokes for different folks, of course. Val likes to occupy herself with digital word games or, if she happens to be outside, for instance yesterday when for 25 minutes nobody had come to unlock the fishes research building where she volunteers, she may snap a few cool photos of handily close spiders or insects for her growing collection of newly identified species.
Many these days, especially the younger set, will use their waiting times for essential phone or text messaging: "Yes. Hello. I'm waiting now."
A more common way to while away the waiting among those a trifle older may be doing crossword puzzles.
Besides the ever popular worrying about how the appointment will go, any number of potential mind games or activities work for me at such times:
Another option I have been using lately is to simply practice building my powers of observation. Moment by moment, life presents a kaleidoscopic succession of 
interesting, absorbing, beautiful, or curious vignettes for one's musing or amusement.
People watching can be more engaging by far than daytime television soap operas.
Animals in one's field of view are frequently most entertaining.
I sometimes like to just close my eyes and meditate for a few minutes. Thousands of books have been written about this kind of activity, most of them probably while their authors were waiting to do something else. I particularly enjoy doing this when tired. With little effort at all, one can concentrate the mind then, and go right to sleep.
Alternatively, one may focus the attention, for instance, on the inhalation or exhalation of the breath (or, if more concrete, on the rising and falling of the belly) or on any sound one hears, even traffic noises.
After doing this for even a few seconds one will, chances are, find anything else of great interest. In a typical 10-minute meditation session while waiting to be called to see a dentist, I began by noticing the sound of a drill boring into a patient's head and then had a series of edifying mentations:
However, one need not follow any such rational techniques for successfully meditating. When all else fails, I meditate like the great Indian yogis do, simply repeating: "Um...um...um (What the heck am I meditating on?)...um...um..." This also works well if one has a stuttering problem.
It may be that none of these methods of getting through waiting time really grabs you. For such truly tough cases, I suggest the best diversion I have managed yet: writing a newsletter essay, perhaps one on waiting!
Whatever works in your case, I do wish you a lifetime of most rewarding waiting.