This excerpt is taken from Sheikh Farooq Rafi Sahib's book Qurbani, Aqiqa, and Ashra Zil-Hijjah.
If it is difficult to find a two-tusked animal among a camel, cow, sheep, or goat, then in cases of hardship, difficulty, and compulsion, the sacrifice of a sheep's khairah is permissible. The evidences are the following hadiths:
① It is narrated from Sayyiduna Jabir (may Allah be pleased with him) that the Messenger of Allah (peace be upon him) said:
لا تذبحوا إلا مسنة، إلا أن يعسر عليكم، فتذبحوا جذعة من الضأن
"You should only slaughter a two-tusked (animal) for sacrifice, but if it becomes difficult for you to get a two-tusked one, then slaughter a sheep's khairah."
② Kalib bin Shahab (may Allah have mercy on him) narrates:
كنا فى سفر فحضر الأضحى فجعل الرجل منا يشتري المسنة بالجذعتين والثلاثة، فقال لنا رجل من مزينة، كنا مع رسول الله صلى الله عليه وسلم فى سفر فحضر هذا اليوم، فجعل الرجل يطلب المسنة بالجذعتين والثلاثة فقال رسول الله صلى الله عليه وسلم: إن الجذع يوفي مما يوفي منه الثني
"We were traveling when Eid al-Adha arrived, so each of us started buying two or three sheep's khairahs in exchange for a two-tusked (goat). Then a man from the Mazinah tribe said that we were traveling with the Messenger of Allah (peace be upon him) when this day (Eid al-Adha) came, and everyone began asking for two-tusked (goats) in exchange for two or three sheep's khairahs. The Messenger of Allah (peace be upon him) said: Indeed, the sheep's khairah suffices for the purpose that two-tusked (goats) fulfill."
Reference: Hasan: Sunan an-Nasa'i, Kitab al-Dahaya, Bab al-Mussannah wal-Jadh'ah: 4388. Mustadrak Hakim: 226/4. Asim bin Kalib and Kalib bin Shahab are truthful narrators.
③ Kalib bin Shahab (may Allah have mercy on him) narrates that a man said:
كنا مع النبى صلى الله عليه وسلم قبل الأضحى بيومين نعطي الجذعتين بالثنية فقال رسول الله صلى الله عليه وسلم: إن الجذعة تجزي ما تجزي منه الثنية
"We were with the Prophet (peace be upon him) and two days before Eid al-Adha, we used to give two sheep's khairahs in exchange for a two-tusked (goat). The Messenger of Allah (peace be upon him) said: Indeed, the sheep's khairah suffices for the sacrifice for which two two-tusked (goats) suffice."
Reference: Hasan: Sunan an-Nasa'i, Kitab al-Dahaya, Bab al-Mussannah wal-Jadh'ah 4389. Sunan Bayhaqi 271/9. Musnad Ahmad 368/5
Benefits:
① If a two-tusked animal is not easily available, sacrificing a sheep or goat is permissible.
② There are two situations when a two-tusked animal is not available:
① Two-tusked animals are rare in the market.
② The price of two-tusked animals in the market is extremely high.
Otherwise, if a two-tusked animal is easily available, sacrificing a sheep or goat is not permissible.
If a two-tusked animal is easily available:
If two two-tusked animals are easily available, then the sacrifice of a sheep's shoulder is not permissible. The sacrifice of a sheep's shoulder is only permissible when obtaining a two-tusked animal is difficult. An ordinary sheep is not sufficient. Also, the narrations regarding the absolute permissibility of the sheep's shoulder are weak and unauthentic.
① It is narrated from Sayyiduna Abu Hurairah (may Allah be pleased with him) that the Messenger of Allah (peace be upon him) said:
نعم أو نعمت الأضحية الجذع من الضأن، فانتهبها الناس
"The best sacrifice is the shoulder of a sheep." (Upon hearing this, people quickly started buying the sheep's shoulder.)
Reference: Weak: Jami` at-Tirmidhi, Chapters of Sacrifices, Chapter on the Trunk of Sacrifices: 1499. Musnad Ahmad: 444/2. Sunan Bayhaqi: 271/9, Kadam bin Abdur Rahman As-Salmi and Abu Kahash As-Salmi are unknown narrators
② It is narrated from Sayyiduna Bilal bin Abi Hilal (may Allah be pleased with him) that the Messenger of Allah (peace be upon him) said:
يجوز الجذع من الضأن أضحية
"The sacrifice of a sheep's shoulder is permissible."
Reference: Weak: Musnad Ahmad 368/6. Sunan Ibn Majah, Book of Sacrifices, Chapter on What is Sufficient for Sacrifices: 3139. Sunan Bayhaqi: 271/9 Muhammad bin Abi Yahya Aslami's mother is unknown.
③ Sayyiduna Abu Hurairah (may Allah be pleased with him) narrates:
نزل جبريل عليه السلام إلى النبى صلى الله عليه وسلم فقال: كيف رأيت عيدنا؟ فقال: لقد تباهى به أهل السماء، اعلم يا محمد! أن الجذع من الضأن خير من السيد من المعز، وأن الجذع خير من السيد من البقر، وأن الجذع من الضأن خير من السيد من الإبل، ولو علم الله ذبحا خيرا منه فدى به إبراهيم عليه السلام
"When Jibra'il (Gabriel) came to the Prophet (peace be upon him), he asked him: Tell me about our Eid (what do you consider its status?). He said: On Eid, the angels of the heavens boast among themselves. O Muhammad! Know that the shoulder of a sheep is better than two tusks of a goat, the trunk of a sheep is superior to two tusks of a cow, and the trunk of a sheep is also better than two tusks of a camel. Because if Allah had considered any sacrifice better than this (sheep's shoulder), He would have given that animal as ransom to Sayyiduna Ibrahim (peace be upon him)."
Reference: Weak: Mustadrak Hakim: 222/4, 233. Musnad Bazzar: 1207. Ishaq bin Ibrahim Hanini is a weak narrator
④ It is narrated from Sayyiduna Abu Hurairah (may Allah be pleased with him) that the Messenger of Allah (peace be upon him) said:
الجذع من الضأن خير من السيد من المعز
"The shoulder of a sheep is better than two tusks of a goat."
Reference: Weak: Musnad Ahmad: 402/2. Mustadrak Hakim 227/4. Ishaq bin Ibrahim Hanini is weak and Thamamah bin Wa'il bin Husain Abu Thufal is an unknown narrator