Praise be to Allah.
There is no saheeh hadeeth which singles out any day or night of the week for praying qiyaam or offering naafil prayers. Everything that has been narrated concerning that is false and is a fabrication against the Messenger of Allaah (peace and blessings of Allaah be upon him), and it is not permissible to narrate it, let alone act upon it. The one who acts upon such false ahaadeeth is introducing an innovation into the religion that is not a part of it, so he should beware of the wrath and punishment of Allaah.
Shaykh al-Islam Ibn Taymiyah (may Allaah be pleased with him) said in Majmoo’ al-Fataawa (23/134):
With regard to prayers on Sunday and Monday and other days of the week – even though it is mentioned by some of those who wrote about raqaa’iq (heart softeners) – there is no difference of opinion among the scholars of hadeeth that all of these ahaadeeth are fabricated and none of the imams regarded these prayers as mustahabb. In Saheeh Muslim it is narrated from Abu Hurayrah (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: “Do not single out the night before Friday for praying or the day of Friday for fasting.” And Allaah knows best. End quote.
Ibn al-Qayyim aid in al-Manaar al-Muneef (95):
The ahaadeeth about praying on certain days or nights, such as praying on Sunday or the night before Sunday, or on Monday or the night before Monday, and other days of the week, are all false ahaadeeth. End quote.
Al-‘Iraaqi (may Allaah have mercy on him) said in Takhreej al-Ihya’ (1/259):
There is no saheeh hadeeth about the days or nights of the week, all of the reports are da’eef (weak) and false. End quote.
Al-Shawkaani said in al-Fawaa’id al-Majmoo’ah (46):
It says in al-Mukhtasar: There is no saheeh report about the prayers during the week. End quote.
There are verses and saheeh ahaadeeth which speak of the virtue of praying qiyaam al-layl in general terms. Please see the answer to question no. 50070.
And Allaah knows best.