This excerpt is taken from Sheikh Farooq Rafi Sahib's book Qurbani, Aqiqah, and Ashra Zil-Hijjah.
The sacrifice of a goat kid is not permissible under any circumstances because the sacrifice requires the camel, cow, or goat to be mature (having two teeth).
However, the Prophet Muhammad (peace be upon him) granted permission to a few specific companions (may Allah be pleased with them) to sacrifice a goat kid. This permission was exclusive to those companions, and the rest of the Ummah is not included in this permission, nor is it correct to generalize this permission. The evidences are as follows:
① Sayyiduna Bara' ibn Azib (may Allah be pleased with him) narrates:
ضحى خال لي يقال له أبو بردة، قبل الصلاة، فقال له رسول الله صلى الله عليه وسلم: شاتك شاة لحم، فقال: يا رسول الله! إن عندي داجنا جذعة من المعز، قال: اذبحها ولا تصلح لغيرك
"My maternal uncle Abu Burda (may Allah be pleased with him) sacrificed before the Eid prayer, and the Messenger of Allah (peace be upon him) said to him: 'Your goat is a meat goat' (meaning its meat is beneficial but it is not valid for sacrifice). He said: 'O Messenger of Allah (peace be upon him)! I have a pet goat kid.' The Prophet (peace be upon him) said: 'Slaughter it, but it is only valid for you and no one else.'"
Reference: Sahih Bukhari, Book of Sacrifices, Chapter: The Prophet’s statement to Abu Burda, "He slaughtered a kid from the goat and it will not suffice for anyone after you": 5556. Sahih Muslim, Book of Sacrifices, Chapter: Its time: 1961. Sunan Abi Dawood, Book of Sacrifices, Chapter: What is permissible in sacrifices from the teeth: 2801
② Sayyiduna Uqbah ibn Amir Al-Juhani (may Allah be pleased with him) narrates:
أن النبى صلى الله عليه وسلم أعطاه غنما يقسمها على صحابته ضحايا، فبقي عتود فذكره للنبي صلى الله عليه وسلم فقال: ضح به أنت
Undoubtedly, the Prophet Muhammad (peace be upon him) gave him (Uqbah ibn Amir) some goats to distribute as sacrifices among the companions of the Prophet (peace be upon him). Then one goat kid remained, and when he mentioned it to the Prophet (peace be upon him), he said: "Sacrifice it."
Reference: Sahih Bukhari, Book of Sacrifices, Chapter: The Prophet’s sacrifice of two rams: 5555. Sahih Muslim, Book of Sacrifices, Chapter: Regarding the sacrifice: 1965. Jami` at-Tirmidhi, Chapters on Sacrifices, Chapter: What has been narrated about the kid from sheep in sacrifices: 1500. Sunan an-Nasa’i, Book of Sacrifices, Chapter: The young and the kid: 4384. Sunan Ibn Majah, Book of Sacrifices, Chapter: What suffices from sacrifices: 3138
③ The following narration from Sunan Bayhaqi is evidence that this permission for sacrificing a goat kid was exclusive to Sayyiduna Uqbah ibn Amir (may Allah be pleased with him) and no one else was permitted to sacrifice a goat kid. Sayyiduna Uqbah ibn Amir (may Allah be pleased with him) narrates:
أعطاني رسول الله صلى الله عليه وسلم غنما أقسمها ضحايا بين أصحابي فبقي عتود منها قال: ضح بها أنت ولا أرخص لأحد فيها بعدك
"The Messenger of Allah (peace be upon him) gave me some goats to distribute among my companions for sacrifice. After distribution, one goat kid remained, so the Prophet (peace be upon him) said: 'Sacrifice it, but I do not permit anyone else to sacrifice a goat kid.'"
Reference: Authentic: Sunan Bayhaqi: 270/9
After narrating this hadith, Imam Bayhaqi (may Allah have mercy on him) states:
إن كانت هذه الزيادة محفوظة كان هذا رخصة لعقبة كما رخص لأبي بردة
"If these additional words (that the permission for the goat kid was exclusive to Sayyiduna Uqbah ibn Amir) are authentic, then this permission for the goat kid is exclusive to Sayyiduna Uqbah (may Allah be pleased with him) just as it was exclusively granted to Sayyiduna Abu Burda (may Allah be pleased with him)."
Reference: Authentic: Sunan Bayhaqi: 270/9
Therefore, the words of the mentioned hadith are authentic and established, and according to Imam Bayhaqi (may Allah have mercy on him), this permission was exclusive to these two companions (Sayyiduna Uqbah ibn Amir and Sayyiduna Abu Burda, may Allah be pleased with them), and no ordinary member of the Ummah is allowed to sacrifice a goat kid.
Benefits:
① Imam Tirmidhi, may Allah have mercy on him, after narrating the hadith of Bara bin Azib, may Allah be pleased with him, writes:
وقد أجمع أهل العلم أن لا يجزي الجذع من المعز وقالوا: إنما يجزئ الجذع من الضأن
"The scholars are unanimous on this issue that the stump (kheera) of a goat is not sufficient for sacrifice, and (in case two-horned animals are not available for sacrifice) only the stump of a sheep is sufficient."
Reference: Jami' Tirmidhi under Hadith: 1508
② Imam Nawawi, may Allah have mercy on him, states:
قال البيهقي وسائر أصحابنا وغيرهم، كانت هذه رخصة لعقبة بن عامر كما كان مثلها رخصة لأبي بردة بن نيار
"Al-Bayhaqi, all the Shafi'is and other scholars say that this (permission regarding the stump of a goat) was specific for Sayyiduna Uqbah bin Amir, may Allah be pleased with him, and Sayyiduna Abu Burda bin Niyar, may Allah be pleased with him."
Reference: Sharh al-Nawawi: 1913
③ Hafiz Ibn Hajar, may Allah have mercy on him, writes:
وفي الحديث أن الجذع من المعز لا يجزي وهو قول الجمهور
"The hadith of Bara states that the sacrifice of the stump of a goat is not sufficient, and this is also the position of the majority of scholars."
Reference: Fath al-Bari: 20/10