Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
We are not aware of any Hadith proving that the Prophet ever performed the Tasbeeh prayer. There is only one Hadith which establishes this prayer.
The Muhaddiths (experts of the Hadith) have different opinions concerning the authenticity of the Hadith of Tasbeeh prayer. However, some of them grade it as Sahih (sound) such as Imams al-Hakim, Ibn Abu Dawood, Ibn Hajar and others. Imam al-Albani also grades it as sound. The text of the Hadith is as follows.
Ibn Abbas reports from the Prophet that The Apostle of Allah (peace be upon him) said to al-Abbas Ibn Abdul Muttalib: “Abbas, my uncle, shall I not give you, shall I not present to you, shall I not donate to you, shall I not produce for you ten things? If you act upon them, Allah will forgive you your sins, first and last, old and new, involuntary and voluntary, small and great, secret and open. These are the ten things: you should pray four Rak’ahs, reciting in each one Fatihat al-Kitab and a Surah. When you finish the recitation of the first Rak’ah you should say fifteen times while standing: “Subhanallah wal-Hamdulillah wa la Ilaha Illa Allah” (Glory be to Allah, Praise be to Allah, There is no god but Allah, Allah is most great). Then you should bow and say it ten times while bowing. Then you should raise your head after bowing and say it ten times. Then you should kneel down in prostration and say it ten times while prostrating yourself. Then you should raise your head after prostration and say it ten times. Then you should prostrate yourself and say it ten times. Then you should raise your head after prostrating and say it ten times in every Rak’ah. You should do that in four Rak’ah. If you can observe it once daily, do so; if not, then once weekly; if not, then once a month; if not, then once a year; if not, then once in your lifetime”.
Allah knows best.