Below is my personal translation of a relevant section from Zuhayli's book الفقه الإسلامي وأدلته. Note that the translation is paraphrased, and may contain mistakes and omissions.
اتفق الفقهاء على أن الشاة والمعز لا تجوز أضحيتهما إلا عن واحد، وتجزئ البدنة أو البقرة عن سبعة أشخاص، لحديث جابر: «نحرنا مع رسول الله صلّى الله عليه وسلم بالحديبية: البدنة عن سبعة والبقرة عن سبعة»۔
وفي لفظ مسلم: «خرجنا مع رسول الله صلّى الله عليه وسلم مهلين بالحج، فأمرنا رسول الله صلّى الله عليه وسلم أن نشترك في الإبل، والبقر، كل سبعة منا في بدنة»۔
The Jurists are agreed that sacrifice of a sheep or a goat is for one. And the sacrifice of a cow or a camel is permitted for seven people., because of the hadith of Jabir: "We sacrificed with the Messenger of Allah at Hudaibiya. A camel for seven and a Cow for seven."
And the wording of Muslim is: "We set out in the state of Ihram for Hajj along, with Allah's Messenger (ﷺ). He commanded us that seven persons should join in a camel and a cow for offering sacrifice."
وأجاز الحنابلة أن يذبح الرجل عن أهل بيته شاة واحدة، أو بقرة، أو بدنة، عملاً بما رواه مسلم عن عائشة أن النبي صلّى الله عليه وسلم ضحى بكبش عن محمد وآل محمد، وضحى بكبشين أملحين أقرنين، أحدهما عن محمد وأمته ، وروى ابن ماجه والترمذي وصححه عن أبي أيوب: «كان الرجل في عهد النبي صلّى الله عليه وسلم يضحي بالشاة عنه، وعن أهل بيته، فيأكلون، ويطعمون ... »۔
[Hanbali]: It is permitted near the Hanbalis that a person sacrifices on behalf of the people of his house, one sheep or a cow or a camel.
Based on the narration of Muslim from Aisha that the Prophet ﷺ sacrificed a ram on his own behalf and on the behalf of his family. And he sacrificed two white rams with horns. One of them on behalf of Muhammad ﷺ and his Ummah.
Ibn Majah and Tirmizi have narrated it from Abu Ayub, and Tirmizi has classed it Sahih that: "At the time of the Prophet (ﷺ), a man would sacrifice a sheep on behalf of himself and the members of his household"
وكذلك أجاز المالكية أن يذبح الرجل الكبش أو البقرة أو البدنة مضحياً، عن نفسه وعن أهل بيته، ولو زادوا عن سبعة إذا كان الاشتراك في الثواب قبل الذبح بشروط ثلاثة: أن يكون قريباً له، ينفق عليه، وساكناً معه، وقد بينتها في شروط صحة الأضحية.
[Maliki]: The Mailis too consider it permissible for a person to sacrifice a ram, or a cow or a camel, on behalf of oneself and the people of his house, even if the shares exceed seven, provided that the sharing is in the reward (and not the price) and three conditions are fulfilled:
- the person offering the sacrifice must be closely related to them
- the person must be spending on them
- the person must live with them
And this has already been mentioned in the conditions of a correct sacrifice.
وقال الشافعية أيضاً: تضحية واحد من أهل البيت تحصل به سنة الكفاية، وإن لم يصدر من بقيتهم إذن
[Shafi]:The saying of the Shafis is the same, that one sacrifice suffices for the people of the house and the sunnah kifayah is fulfilled, even if it is done without their permission.
My understanding of the Hanafi position is that it is an individual obligation on everyone who is able, similar to prayer, fasting and hajj. So every member of a household on whom the obligation applies must offer an individual sacrifice and they can not waive this obligation by sharing in a sheep or goat.
Evidence cited by the Hanafis includes this hadith:
من كان له سعة ولم يضح فلا يقربن مصلانا
Whoever can afford it, but does not offer a sacrifice, let him not come near our prayer place.
Sunan Ibn Majah