In a particular month of some year, there are three Mondays which have even dates. On which day of the week does the 15th of that month fall?

Dates of a specific weekday in any month always repeat after intervals of $7$ days. Therefore, the dates alternate between odd and even integers.
For a month to contain exactly three Mondays with even dates, there must be a total of $5$ Mondays in that month, where the first, third, and fifth Mondays land on even dates.

Let the first Monday be on an even date $d$.
The sequential dates of all five Mondays will be:

$$d,d+7,d+14,d+21,d+28$$

For $d$ to be even and $d+28\le 31$, the only possible value for $d$ is $2$.
Therefore, the five Mondays of that month fall on the $2\text{nd}$, $9\text{th}$, $16\text{th}$, $23\text{rd}$, and $30\text{th}$.

If the $16\text{th}$ is a Monday, then the $15\text{th}$ must be the preceding day, which is a Sunday.