There is a variety of threads on this topic in Tech Log .. one you might start with is
here.
In general,
(a) for a specific Type, the OEM TP folk have spent a lot of time looking at OEI operations and the appropriate Type techniques for the pilots are specified in the AFM, FCOM, etc.
(b) for generic thinking ..
First, we have the dynamic bits where hands get sweaty for a little while ..
(i) failure creates an undesired yaw .. the closer to Vmc the more dramatic the dynamics will be. Keep in mind that things happen very, and increasingly, rapidly in the last few knots coming back to Vmc.
(ii) yaw is controlled with rudder (roll is controlled with aileron). The more dynamic the situation, the more rapidly the pilot has to do whatever may be required.
then we settle down into some sort of quasi-static continued flight situation ..
(iii) IF we end up wings level, then there will be a residual out of balance force in that the rudder causes a lateral force and acceleration toward the dead side. This results in a sideways velocity with gives a small sideslip angle and, generally, reduced climb or (increased descent) rate. There will be some minimum IAS which can be achieved in this configuration.
(iv) if we are clever, we can bank a little into the live side and generate an opposite direction slip to balance the rudder-induced slip. To keep everyone honest and to keep the aircraft away from too much slip toward the live side, the Standards impose a maximum bank into the live side of 5 degrees. Unless the AFM says different, it is reasonable to presume that controlled flight at Vmc will require this bank angle. We also find that, with a bit of bank, we don't need as much rudder as the slip into the live side also generates a stabilising yawing moment .. ergo, we can slow down a bit by using a bit more rudder and achieve a lower speed. The AFM Vmca speed usually will be for the 5 degree bank into the live side configuration.
(v) while we are back near Vmc (which we really should avoid like the plague) we are interested in control and staying blue side up - not performance.
(vi) once we can get to a speed margin somewhat above Vmc, the control problems are reduced greatly and we can concentrate on performance .. however abysmal that may be in the real world.
(vii) generally we will get best climb capability somewhere near zero slip (think about it .. how do you increase your ROD on final once you are configured to full flap, gear down, and throttles closed ?). Generally this will occur for a bank angle into the live side of around 2-3 degrees.
(viii) I don't know what your thoughts are but, for me, flying 2-3 degrees wing down is hard work (and inappropriate for some AH systems due to re-erection considerations).
Considering that
(a) climb penalties tend to be similar for wings level and 5 degrees
(b) climb improvement around 2-3 degrees is not incredibly dramatic
(c) jets sometimes have spoiler considerations with which to contend
we generally end up flying wings level for the post failure continued flight scenario .. ie after the dynamic things have been sorted out.