Praise be to Allah.
What is proven from the Prophet (peace and blessings of Allaah be upon him) is that he used to wipe his ears, not wash them. He would wipe the outer part with his thumbs and the inner part with his forefingers, with whatever water was left on his hands after wiping his head.
Al-Tirmidhi (36) narrated from Ibn ‘Abbaas (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) wiped his head and ears, inside and outside. Abu ‘Eesa al-Tirmidhi said: The hadeeth of Ibn ‘Abbaas is a hasan saheeh hadeeth, and this is what should be done according to most of the scholars, who think that the ears should be wiped, inside and outside.
Al-Nasaa’i (74) narrated that Ibn ‘Abbaas (may Allaah be pleased with him) said, describing the wudoo’ of the Prophet (peace and blessings of Allaah be upon him): Then he wiped his head and his ears, the inside with his forefingers and the outside with his thumbs. Classed as saheeh by al-Albaani in Saheeh al-Nasaa’i.
In Tuhfat al-Ahwadhi it says:
The outside of the ear is the part that is closest to the head and the inside is that which is closest to the face. End quote.
Abu Dawood narrated that al-Miqdaad ibn Ma’di Yakrib (may Allaah be pleased with him) said: I saw the Messenger of Allaah (peace and blessings of Allaah be upon him) doing wudoo’ and when he reached the wiping of the head, he placed his hands on the front of his head and moved them until he reached the back of his head, then he brought them back to the place where he started, and he wiped his ears, inside and outside, and put his fingers in his ears. Classed as saheeh by al-Albaani in Saheeh Abi Dawood.
al-Nawawi said in al-Majmoo’ (1/443):
With regard to the way in which the ears are wiped, Imam al-Haramayn, al-Ghazaali and others said: He should take water in his hand and put his forefingers inside his ears and rotate them, and pass his thumbs over the outside of his ears. End quote.
If there is any waxy substance over the opening of his ears, it should be removed, unlike that which is inside, as it is not prescribed for water to enter the ear, rather what is required is to wipe them, not wash them, as stated above.
And Allaah knows best.