Consider the following statements:

1. The minimum number of points of intersection of a square and a circle is 2.

2. The maximum number of points of intersection of a square and a circle is 8.

Which of the above statements is/are correct?

The correct option is 2 only.

Let us analyze both statements step-by-step to determine their correctness.

Analysis of Statement 1:
Statement 1 asserts that "The minimum number of points of intersection of a square and a circle is 2."
If a circle and a square do not touch or cross each other at all (for example, if the circle lies entirely outside the square without touching, or entirely inside the square without touching), the number of points of intersection is 0.
Even if we consider geometric figures that do intersect, a circle can be tangent to a square (touching at exactly 1 point). Therefore, the minimum number of points of intersection is not 2.
Hence, Statement 1 is incorrect.

Analysis of Statement 2:
Statement 2 asserts that "The maximum number of points of intersection of a square and a circle is 8."
A square consists of 4 straight line segments (its edges). A straight line can intersect a circle in at most 2 points.
Since the square has 4 distinct side segments, each side segment can intersect the circle in at most 2 points.

$\text{Maximum points of intersection}=4\times 2=8$

This occurs when the circle crosses each of the four sides of the square twice (for instance, when the center of the circle is located near the center of the square and its radius is larger than the distance from the center to the sides, but smaller than the distance from the center to the corners).
Therefore, the maximum number of intersection points between a square and a circle is indeed 8.
Hence, Statement 2 is correct.

Conclusion:
Since Statement 1 is incorrect and Statement 2 is correct, the correct answer is 2 only.