What is the rationale for single engine military aircraft?
Because single engine fighters are substantially cheaper to purchase and operate.  Exact figures are hard to obtain, but as an example, an F-15 squadron will spend about 25,000 USD per flight hour whereas an F-16 Squadron spends about 15,000 USD per flight hour.   Effectively, you can purchase and operate twice the fighter force with single engine fighters than with multi-engine fighters.   This becomes very attractive for land based air forces belonging to countries with small defense budgets to operate the maximum number of combat aircraft per unit of currency spent. 

Another major advantage to single engine fighters is maximized performance in a minimal size and weight package.  Adding a second engine will produce more thrust but at the cost of greater weight and fuel consumption.  This in turn requires more fuel be carried with a heavier airframe, creating a vicious circle for the design of these airplanes.  Typically in fighter design you want a specific weight ratio $W_a = W_{af}/(W_{af} + W_f)$ as low as possible.  This translates into an airplane which is a nimble, high capacity gas can with as low an empty weight as possible for a given range profile.  This is extraordinary difficult to do with a multi engine airframe.

The major advantage to a multi engine fighter is the same advantage that any other multi engine airplane has: they can carry larger payloads with greater system redundancy.   The large airframes accommodate big powerful fire control radars for detecting and tracking targets at very great distances, etc.  Certain units, particularly the United States Navy, prefer multi engine airplanes for overwater operations for the redundancy of having an additional engine to get you back home.   So too is this a design feature of the A-10 attack jet which anticipates enduring battle damage and the additional engine offers a better chance of limping it home if it gets shot up.

While the redundancy of two engines may seem ideal for combat ops, keep in mind an engine failure combined with the compressor and boat tail drag from the dead engine results in an airplane which has lost 50% of its total available engine thrust and 80% of its combat effectiveness.  All you could do after an engine failure would be to limp the airplane home to a friendly base - provided the enemy is gracious enough to let you do so without jumping your six and blowing you out of the sky!