Many Muslims, after burping or sighing, systematically say: "Astaghfirullâh". It is my understanding, though, that there's no such sunnah and that it is merely cultural.
Sheikh `Abd al-Rahmân al-`Ajlân, lecturer at the Grand Mosque in Mecca
It is permissible to seek forgiveness at any time. However, a person should not make a specific habit out of doing so after burping or sighing, because this is not according to the Sunnah of our Prophet (peace be upon him) and some people might misconstrue it to be a Sunnah.
Source: Islam Today
-- Al Arabiya Digital