Each Flap setting will have it's own Vx and Vy. Both are related to Lift and DRAG, with the higher flap settings having the lower Vx/Vy, with Vx/Vy increasing as flap setting decreases. The actual Angle / Rate of Climb will be decreased in all cases when any flap is used.
I know of no aircraft where a flap setting caused less, or even equal drag than the clean configuration, thus the best Angle / Rate of Climb will always be achieved with zero flap at the appropriate Vx/Vy. If one such aircraft does exist, why not leave the flaps out permanently?
There are many instances where pilots conduct their initial climb, often fairly lengthy, with Takeoff Flap selected and at Vx for that flap setting. This will be so where critical obstacles are present. Although the climb angle would be improved in the clean configuration at the clean Vx, it necessitates an acceleration from the ''flapped' Vx to the clean Vx. The required acceleration could only be accomplished at a REDUCED climb angle, thus, the overall climb gradient would be less at a time when obstacles are critical.
Once clear of obstacles, climb angle may be reduced, allowing acceleration to the 'Flaps Up' speed.
The same principal applies equally to a Cessna 172 as for a Boeing 747.
Hope that that clarifies things a bit.
Old Smokey