Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
You are not obliged to pay the Zakah on your monthly salary if it is less than Nisaab (minimum amount of property liable to Zakah that is 85 gram of pure gold or its equivalent). If it reaches Nisaab and one lunar year has passed over it and remain unused or consumed then you have to pay its Zakah. The amount of Zakah you have to pay is 2.5 percent of your whole wealth. If one saves something from his salary and it reaches Nisaab, after completing the Nisaab if a lunar year has passed on that amount then he has to pay its Zakah. If the amount of Nisaab reduces during the year and he regains it, then he can resume the year for paying Zakah from the time of regaining the Nisaab. You are obligated to pay the Zakah of your wealth since you are the owner of that money. Your wife is not required to pay Zakah on the share you give her. But if she has her separate sources of income, such as salary or whatever then she has to pay Zakah with the two conditions mentioned above. Allah knows best.