For arbitrary systems of charges it is not possible to integrate the differential equations of a line of force. Equipotentials always have non-local equations (from definition), but then it is difficult to say anything about their shape in the general case; the conclusion that points where E = 0 lead are centers of separation between families of equipotentials holds, of course. One can easily get an overall picture of the field with the aid of Field Expert, write a qualitative description, and then focus on "hot points" to compute potential and field strength rigorously. Section 2.1 proves the effectiveness of this method.
The general behavior of the field is the following:
NOTE: multipole configurations are rather hard to obtain, requiring very precise geometry; the quadripole may be the easiest -- four charges of equal magnitude and alternating signs in the corners of a square.
Another aspect from which I have shied away up to now is that the images produced by Field Expert are intriguing and, well, beautiful. I take the freedom to include some examples.