Let the initial salary of the person be $100$.

Given that the person saves $10\%$ of his salary:

• Initial saving = $10\%$ of $100 = 10$
• Initial expenditure = Initial salary - Initial saving = $100 - 10 = 90$

Now, the salary increases by $12\%$ and the expenditure increases by $10\%$:

• New salary = $100 + 12\%$ of $100 = 100 + 12 = 112$
• New expenditure = $90 + 10\%$ of $90 = 90 + 9 = 99$

The new saving will be the difference between the new salary and the new expenditure:

• New saving = New salary - New expenditure = $112 - 99 = 13$

Now, we calculate the change in his saving:

• Increase in saving = New saving - Initial saving = $13 - 10 = 3$
• Percentage increase in saving = $\left( \frac{\text{Increase in saving}}{\text{Initial saving}} \right) \times 100\%$
• Percentage increase = $\left( \frac{3}{10} \right) \times 100\% = 30\%$

Therefore, there is a $30\%$ increase in his saving per month.