Praise be to Allah.
General ruling on food and drinks
The basic principle with regard to food and drink is that it is permissible unless there is proof that it is haram. Allah says (interpretation of the meaning): “He it is Who created for you all that is on earth.” [al-Baqarah 2:29]
Ibn 'Abbas said:
“The people of the Jahiliyyah used to eat some things and refrain from others because they found them off-putting. Then Allah, may He be exalted, sent His Prophet (blessings and peace of Allah be upon him), revealed His Book, and explained what is permissible and what is forbidden. Whatever the Quran has permitted is permissible and whatever it has forbidden is forbidden, and whatever it does not mention is pardoned. Then he recited: “Say (O Muhammad): I find not in that which has been revealed to me anything forbidden to be eaten by one who wishes to eat it, unless…” [al-An‘am 6:145] until the end of the verse.” (Narrated by Abu Dawعd (3306); classed as sahهh by Shaykh al-Albani (may Allah have mercy on him).
Al-Hafiz Ibn Hajar (may Allah have mercy on him) said:
“The second type is that concerning which no prohibition was narrated; it is permissible, but on condition that it be slaughtered properly, such as duck and waterfowl.” (Fath al-Bari)
Are duck and pigeons halal?
With regard to ducks and pigeons, there is no evidence to suggest that it is haram to eat them, so we go back to the original principle, which is that they are permissible.
In fact the permissibility of eating pigeons can be seen in the fact that the Sahabah (may Allah be pleased with them) ruled that if a muhrim (pilgrim in ihram) hunted the pigeons of the Haram (sanctuary), the expiation for that is a sheep, which indicates that it is permissible to eat them (i.e., pigeons -- the prohibition is on hunting when in ihram and within the boundaries of the Haram).
Ibn Qudamah (may Allah have mercy on him) said:
“Similar rulings were made by ‘Umar, ‘Uthman, Ibn ‘Umar, Ibn ‘Abbas and Nafi‘ ibn ‘Abd al-Harith…” (Al-Mughni, 3/247)
An-Nawawi (may Allah have mercy on him) said:
“Our companions unanimously agreed that it is permissible to eat ostriches , chickens… ducks, sand grouse, sparrows, larks, francolin and pigeons…” (Sharh al-Muhadhdhab, 7/22)
He also said:
“Whatever lives in the water and on land, including waterfowl such as ducks, geese and so on, is halal as stated above, but if any of these dies of natural causes (i.e., is not slaughtered in the proper manner) it is not halal, and there is no difference of scholarly opinion on this point. Rather it is essential that it be slaughtered properly.” (Sharh al-Muhadhdhab, 9/35)
For more about food and nourishment, please see this section: Food & Nourishment
And Allah knows best.