Praise be to Allah.
The scholars (may Allah have mercy on them) differed as to whether Salat al-Tasbeeh is prescribed in Islam. The reason for this difference of opinion concerning it is the difference of opinion concerning the soundness of the hadeeth which speaks of it. The prominent scholars are of the view that it is da‘eef (weak).
1. Ibn Qudamah (may Allah have mercy on him) said in al-Mughni (1/438):
With regard to Salat al-Tasbeeh, Ahmad said: I do not like it. It was said to him: Why? He said: There is nothing saheeh (authentic) about it, and he waved his hand as if dismissing it. End quote.
2. Al-Nawawi (may Allah have mercy on him) said in al-Majmoo‘ Sharh al-Muhadhdhab (3/547-548):
Al-Qadi Husayn, and the authors of al-Tahdheeb and al-Tatimmah said: Salat al-Tasbeeh is mustahabb (desirable), because of the hadeeth narrated concerning it. But this view of it as mustahabb is subject to further discussion, because the hadeeth in question is da‘eef (weak), and it involves changing the regular format of the prayer, so it should not be done without a hadeeth (to support it). Al-Tirmidhi said: More than one hadeeth was narrated from the Prophet (peace and blessings of Allah be upon him) concerning Salat al-Tasbeeh, but not many of them are saheeh. And al-‘Aqeeli said: There is no proven hadeeth about Salat al-Tasbeeh. End quote.
3. Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said in Majmoo‘ al-Fatawa (11/579):
The best of the reports narrated about these prayers is the hadeeth about Salat al-Tasbeeh, which was narrated by Abu Dawood and al-Tirmidhi, but nevertheless none of the four imams spoke of it; in fact, Ahmad regarded the hadeeth as da‘eef, and these prayers are not mustahabb.
4. Al-Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in Majmoo‘ Fatawa Ibn ‘Uthaymeen (14/327):
What seems more likely to be correct in my opinion is that Salat al-Tasbeeh is not sunnah (from the prophetic teachings), and that the hadeeth about it is da‘eef. This is for several reasons:
(i) The basic principle concerning acts of worship is that they are disallowed unless there is proven evidence that they are prescribed.
(ii) The hadeeth is mudtarib (weak); there is a difference of opinion concerning it for several reasons.
(iii) None of the imams regarded it as mustahabb.
(iv) If this prayer was prescribed, the ummah would have narrated a report concerning which there was no doubt and it would have been well known among them because of its great benefit. Moreover, it is not like any other act of worship, because there is no act of worship in which so much choice is given that it may be done once every day, or once in a week, or once in a month, or once in a year, or once in a lifetime. Because the weak hadeeth states that it is of great benefit and because it is not like any other act of worship, yet it is not well known and was not transmitted, we know that there is no basis for it, because if an act of worship is different from other acts of worship, and if it were of great benefit, people would pay attention to it and transmit it, and it would be popular and well known. Because that is not the case with regard to this prayer, we know that it is not prescribed. Hence none of the imams regarded it as mustahabb, as Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said. Among the nafl (voluntary) acts which are proven to be prescribed in Islam there is a great deal of goodness and barakah (blessing) for the one who wants more, so he should be content with that which is proven, and he has no need of that which is the subject of differences of opinion and doubts. End quote.
For more information, please see the answer to question no. 14320
And Allah knows best.