The Hadith describe the excellence of Salaat Ut Tasbeeh. There is a Hadith in Sunan Ibn Majah:
Translation: It has been narrated on the authority of Hadhrat Abdullah bin Abbas (May Allah Ta’ala be well pleased with them) that the Holy Prophet (Sallallahu alaihi wa sallam) told Hadhrat Abbas bin Abdul Muttalib (May Allah Ta’ala be well pleased with them) that: O Abbas! O My uncle! Should I not grant you? Should I not honor you? Should I not treat you with Ihsaan (kindness)? Should I not tell you those 10 things, which when you perform, Allah (Subhanahu Wa Ta’ala) will forgive all your sins, whether they are of the past or of the future, the new or the old, whether they were committed unwittingly or deliberately, whether they are small or big, whether they were done openly or in secret. Offer 4 Raka’at, then He (Sallallahu alaihi wa sallam) described the manner of offering Salaat Ut Tasbeeh.
(Sunan Ibn Majah, Pg. No. 99)
In Jame’ Tirmidhi, details about offering Shab Bara’at are mentioned on the authority of Hadhrat ‘Abdullah bin Mubarak.
Translation: Make the Niyyah (intention) of offering 4 Raka’at Salaat Ut Tasbeeh. After the first Takbeer (Takbir Tahrima), after Sana, recite 15 times:
Subhanallahi Walhamdulillahi Wa La Ilaha Illallahu Wallahu Akbar
Then recite the Ta’awuz and Tasmiya, then Surah Fatiha and another Surah after it. Then before going into Ruku recite the abovementioned Tasbeeh 10 times. Then perform Ruku and after the Tasbeeh of Ruku recite the given Tasbeeh 10 times, then get up from the Ruku and after saying “Sami’allahu………” recite the Tasbeeh 10 times. Then perform the first Sajdah. After the Tasbeeh of the Sajdah, recite 10 times. Then in the sitting between the Sajdah, recite the Tasbeeh 10 times, then again in the second Sajdah.
When you stand up for the next Raka’at, before Surah Fatiha, recite the Tasbeeh 15 times and continue as said. In each Raka’at, the Tasbeeh will be recited 75 times, in 4 Raka’at 300 times. End the Salaat after 4 Raka’at.
Source : www.ziaislamic.com/english/interface/ArticleUrduDet1.php?ids=610&&key=SALATUL%20TASBEEH