square
Rectangles
Not generally.
Most quadrilaterals do not.
A square and a rectangle have diagonals of the same lengths.
rectangle and parallelogram
rectangle and parallelogram
Most quadrilaterals will not have congruent diagonals: rectangles (incl squares) and isosceles trapezium are the exception.
A square
Parallelograms or isosceles trapezium.
No. An isosceles trapezium (isosceles triangle with its apex removed) would have congruent diagonals but it is not a parallelogram.
square, rectangle and an isosceles trapezoid.