One of the classic asymmetrical curves is the Pareto Curve, which has been popularized as the 80-20 rule, where in management 80% of your problems are caused by 20% of your people; or 20% of your people do 80% of the work. Helpdesk ticket closure times are another good example, where the bulk of the them are closed in a fairly short range of time, but others, especially those requiring equipment replacement, extend the overall time range far to the right. Back when I was at GE Fleet Services, I would check the 3rd quantile, or the mark at which the technician had closed 75% of their tickets, rather than their average time to close.
Looking at a 2d6 distribution we find, 97.22% probability that you roll less than a 12; 91.66% less than an 11; 83.53% less than a 10 and 72.42% less than a 9. We can create a reasonable long tail distribution very simply. Whenever an 11 or 12 is rolled, roll another d6 and add it to 10, producing a 2-16 distribution.
I used anydice to look at the probabilities. Their calculator will allow you to see the percentages for any combination of dice.
I haven't come up with as simple a way to do a long left hand tail, but I don't have a use case either. If I get one I'll probably just flip the scale at the bottom and read from right to left.
As I said, the use case I identified is random NPC levels, I'd be interested in hearing of any other use cases you might come up with.