The answer is: When it makes things simple.
Let's expand that answer a bit. First the things in question are:
(1) The integrand f(x,y,z)
(2) The equations of the surfaces bounding the region
These are likely to simplify in cylindrical if either if the things prominently features sqrt(x^2 + y^2), which is simply r in cylindrical, or sqrt(x^2 + y^2 + z^2), which is simply rho in spherical coordinates.
Another clue is this: one should keep in mind which kind of surfaces will lead to constant limits in each coordinate system. Certainly constant limits are the best kind. The lists are:
Conversely, if for example your region is bounded in part by a plane, say
x + y + z = 0
which is not one of the types listed above, you'd probably better not use either cylindrical or spherical. This same plane in spherical coordinates, for example, has the equation
phi= cot^(-1)(cos(theta) + sin(theta))
which, if it is allowed to appear in your limits of integration, will under most circumstances lead to disaster. Keep it out.
The equation would not be so bad in cylindrical, but that is only because the plane does at least pass through the origin. Consider instead
x + y + z = 1
and translate this to polar and spherical. Yuck. The only good planes if you are using spherical or cylindrical coordinates are those in the lists above.
Next, look at the bounding surfaces involved. If the equations for them prominently feature x^2 + y^2, you're getting a signal to use cylindrical, but maybe also spherical.. If they prominently feature x^2 + y^2 + z^2, you're getting a signal to use spherical.
Moreover, if these surfaces consist of pieces from either of the two lists above, you're getting a signal to use the corresponding set of coordinates.
The problem is that you will often get mixed signals: The integrand may tell you "use spherical", while the bounding surfaces are telling you "use cylindrical". You'll have to decide which advice is better. This is a judgement call you'll have to make, and experience alone can prepare you to judge right. Practice is the only way.
The second question is: O.K., so I've decided to use cylindrcal, say. I've got three variables, so there are 3! = 6 possible orders of integration. How do I know which is best?
And as we've seen, the order of inegration makes a big difference in feasibility of integration. (See also Example 1 below).
The answer here is: By experience, really. But there are some useful things to say, nonetheless.
First, the most important thing to decide is which variable will be integrated last. The reason this is most important, is this choice determines what sort of two dimensional slices we will work with in figuring out our limits. It is generally nicest to have these be planes. They will be if theta is held constant until the last, as we can see from the lists of simple surfaces. If we make this choice, the slice will be the cross section that we can see in a side view diagram. This is easiest to graph, usualy.
Thus, as a general rule, it is often a good idea to hold theta fixed until the last integration for setting up the limits. If you do so, a side view picture, with all of the lines labled by equations will usually contain all the information you need to set up the limits.
If, however, you are working in cylindrical, and save r for the last integral, your slices will be pices of cylinders, as you see on the above list of simple surfaces. There may be cases where this is what you would want to do, especoally with certain integrands. But these pieces are not flat, so drawing your two dimensional picture of a typical slice may be tricky (though you can "unroll" them...) If you were working in spherical, and save rho for last, your slices would be pieces of spheres. These are not really not flat (can't be unrolled), and your two dimensional picture will be quite hard to draw in general.
So the most likely to be simple orders in cylindrical are r, thenz, then theta, or z, then r, then theta.
The most likely to be simple in Spherical is rho, then phi, then theta.
But for certain regions and certain integrands, there will be exceptions.
This is about as much as one can say without getting more into examples. Here some are: