Let $S$ be a set of five points in the plane, no three of which are collinear. We need to determine the minimum possible value of $\frac{M(S)}{m(S)}$ where $M(S)$ and $m(S)$ are the maximum and minimum areas of triangles that can be formed by any three points from $S$.

### Analysis:

1. Configuration of Points:
Consider placing the five points of the set $S$ as the vertices of a convex pentagon. Given that no three points are collinear, any three points will form a triangle.

2. Area Consideration of Triangles:
For a regular pentagon, the area of triangles formed by its diagonals and adjacent sides can have a vast difference. The crucial point here is the relationship of the diagonals and sides in a regular pentagon.

3. Connection to the Golden Ratio:
A regular pentagon comprises triangles and segments that relate to the golden ratio, $\phi = \frac{1+\sqrt{5}}{2}$.

4. Computation of Area Ratios:
In a regular pentagon, the triangle with maximum area can be a diagonal and two adjacent vertices forming the base and height (a kite configuration). At the same time, the triangle with a minimal area could be one of the smaller internal triangles formed by consecutive vertices.

The side length relation in a regular pentagon upholds that the ratio of the diagonal to the side is $\phi$.

5. Outcome:
The maximum triangle is comprised of two adjacent sides and a diagonal. The smallest triangle is one with minimized deviation from the side. The ratio of their areas directly yields our result through proportional scaling references consistent with the golden ratio.

Thus, the minimum possible value of $\frac{M(S)}{m(S)}$ is the golden ratio:
$$\boxed{\phi = \frac{1+\sqrt{5}}{2}}$$

This aligns with the special properties of the geometric arrangement and confirms the constraints provided in the problem.