Since angles \( X \) and \( Y \) are supplementary, their measures add up to \( 180^\circ \). Given that angle \( X \) measures \( 127.42^\circ \) and angle \( Y \) measures \( (m - 12)^\circ \), we can set up the equation:
\[ X + Y = 180^\circ \]
Substituting the values we have:
\[ 127.42 + (m - 12) = 180 \]
Now, simplify the equation:
\[ 127.42 + m - 12 = 180 \]
Combine like terms:
\[ m + 115.42 = 180 \]
To isolate \( m \), subtract \( 115.42 \) from both sides:
\[ m = 180 - 115.42 \]
Calculating the right side:
\[ m = 64.58 \]
Now that we have found \( m \), we can substitute it back to find \( m∠Y \):
\[ Y = m - 12 = 64.58 - 12 = 52.58^\circ \]
Thus, the measure of angle \( Y \) is \( \boxed{52.58^\circ} \).