Generally speaking, we would like to stress that Qiyam Al-Layl (Night Vigil Prayer) is an act of Sunnah offered on all the nights of the year. As for the Tarawih Prayer, it is an act of Sunnah peculiar to the month of Ramadan to be offered starting from the first night of Ramadan. Regarding offering the Tarawih Prayer on the night of the 30th of Sha`ban or laylat ash-shak, on which people expect to sight the crescent of Ramadan, generally, Muslims may not offer it until they make sure the crescent has been sighted.
If astronomers have sighted the crescent moon of Ramadan on that night, people can offer the Tarawih Prayer. If not, there is no Tarawih Prayer to be offered. People may, instead, offer Qiyam Al-Layl, which is a Sunnah throughout the year.
However, there are some scholars who believe that it is desirable for people to offer the Tarawih Prayer on laylat ash-shak [before determining whether or not Ramadan begins the next day]. Anyway, people should not differ a lot regarding that issue, for there is no clear-cut religious evidence on either opinion.
In his book Ahkam Al-Qur’an, Al-Gasas, a Hanifi scholar, said: A lunar month starts from the night in which the crescent of that month appears, and hence, that night is to be regarded as part of the month in question. By analogy, the night on which the crescent of Ramadan is seen is regarded the first night of Ramadan, and so, the Tarawih Prayer is to be offered then. It was reported that the Prophet (peace and blessings be upon him) said: “On the first night of Ramadan devils are chained.” This proves that the beginning of a lunar month is from its very first night.

In his book Al-Mughni, Ibn Qudamah said:

Our followers from the Hanifi School of thought have differed as to the desirability of offering the Tarawih Prayer on laylat ash-shak. It was reported on the authority of Al-Qadi [an eminent Hanifi scholar] that he said: That issue was raised during the lifetime of Sheikh Abu `Abdullah (who was a prominent Hanafi scholar), and he offered the Tarawih Prayer on laylat ash-shak. Qadi Abu Ya`la offered it also on laylat ash-shak. They did so because the Prophet (peace and blessings be upon him) said: “Allah Almighty has ordained you to fast it (Ramadan), and I have made it an act of Sunnah for you to offer an additional Prayer during its nights [i.e., the Tarawih Prayer].”
On the other hand, Abu Hafs Al-`Ukbarai, an eminent Hanifi scholar, would not offer the Tarawih Prayer on laylat ash-shak. In that respect he followed in the footsteps of the Prophet’s Companions and the righteous predecessors. It was not reported that they would offer the Tarwih Prayer on laylat ash-shak.