Is it better to say the tasbih silently or aloud after the prayer?

The first and best approach is to say all dhikrs (remembrances) silently, unless there is an established command to vocalize them. This includes the dhikrs after prayer, as it is closer to sincerity and further from ostentation (riya’) and disturbing others. Some scholars have permitted vocalizing them, citing the hadith of Ibn Abbas that they would only know the prayer had ended by the takbir (utterance of "Allahu Akbar") thereafter. The response to this is that what was meant was the reverberation resulting from the collective dhikr, even when recited softly. In any case, there is no harm for one who vocalizes or recites silently with the intention of following the Sunnah. The soul inclines towards whispering, unless vocalization is for the purpose of teaching the ignorant or in a place where there are common folk.