For the Theravāda, nibbāna is an ultimately real dhamma (paramatthadhamma) and the only dhamma that is not conditioned (asaṅkhata). It is an object of supramundane cognition (lokuttaracitta) and is included in the mental phenomena sensory sphere (dhammāyatana) and the mental phenomena component (dhammadhātu). The four paths, four fruits, and nibbāna are classified as the unincluded level (apariyāpanna bhūmi), that is, not included in the sensual realm, the form realm, or the formless realm. According to the Visuddhimagga, nibbāna "has peace as its characteristic. Its function is not to die; or its function is to comfort. It is manifested as the signless; or it is manifested as non-diversification (nippapañca)."
According to the Sarvāstivāda, nirvāṇa is an analytical cessation (pratisaṃkhyānirodha) that is a disjunction from impure dharmas that occurs through analysis (pratisaṃkhyāna), which is a specific type of discernment (prajñā). This analytical cessation is substantially existent (dravyasat) and ultimately exists (paramārthasat).
For Sautrāntika commentators nirvāṇa as an analytical cessation (pratisaṃkhyānirodha) is a merely a conceptual designation (prajñapti) and doesn't refer to an entity or state that is substantially existent (dravyasat). It is a non-implicative negation (prasajyapratiṣedha), that is, a negation that doesn't imply the presence of some other entity. Therefore nirvāṇa simply refers to a cessation that is the termination of defilements that are abandoned by the correct practice of the noble path.
According to the Yogācāra, for those on the bodhisattva path, nirvāṇa is non-abiding (apratiṣṭha nirvāṇa). The dependent nature (paratantrasvabhāva) is the basis (āśraya) of both defilement and purification. The all-basis consciousness (ālayavijñāna) is the defiled portion (saṃkleśabhāga) of the dependent nature. Purified suchness (viśuddhā tathatā) is the purified portion (vyavadānabhāga) of the dependent nature. Synonyms for purified suchness are the perfected nature (pariniṣpanna) and non-abiding nirvāṇa. Non-abiding nirvāṇa is the revolved basis (āśrayaparāvṛtti) that has eliminated defilements without abandoning saṃsāra.
Madhyamaka authors accept the notion of non-abiding nirvāṇa, but they don't use the three natures model used by the Yogācāra. Rather, they simply consider all things to be conceptual designations (prajñapti) that are empty of nature (svabhāva). For them, conceptual designations are relative truth (saṃvṛtisatya) and only emptiness is ultimate truth (paramārthasatya).
Zen, Pure Land, Vajrayāna, etc., are practice traditions more so than doctrinal schools, and authors writing from any of these perspectives would generally rely on Yogācāra or Madhyamaka śāstras or a specific Mahāyāna sūtra.