To determine on which plane the center of the sphere \( S \) lies, we must first consider the conditions under which the sphere is defined.

The sphere \( S \) passes through the points \( A(1, 0, 0) \), \( B(0, 1, 0) \), and \( C(0, 0, 1) \). The center of this sphere with the smallest radius lies on the plane that is equidistant from these points.

The plane that is equidistant from all these points can be determined by considering the conditions: - A plane in the form \( ax + by + cz + d = 0 \) that equates to being parallel to the vector formed by joining these points.

By symmetry and using the condition of the minimum radius, the center must lie on the plane with equation \( x + y + z = k \).

Since the points \( A(1, 0, 0) \), \( B(0, 1, 0) \), and \( C(0, 0, 1) \) are on the surface of the sphere, the plane must pass through the sphere’s center such that the sum of the coordinates of all these points remains balanced around the center.

By analyzing the sphere through geometry and symmetry, the center must lie on the plane:

x + y + z = 1

Thus, the correct answer is that the center of the sphere \( S \) lies on the plane:

x + y + z - 1 = 0

We can now verify which option matches this plane equation. Comparing to the given options, we see that Option \(x + y + z - 1 = 0\) is indeed the correct solution.

Therefore, the correct choice is:

x + y + z - 1 = 0