Most prominent ritual that people do in shawwal is fasting the six days of shawaal.

You have to make up for the missed days of Ramadan so that you would have fulfilled fasting the whole month of Ramadan. And after that you must… if you wish fast the six days of shawwal in order to fall under the category and under the reward of the Prophet (saw) that afterwards meaning after fasting Ramadan you follow that with six days of shawwal.

if I select to fast Mondays and Thursdays with the intention of the six days of shawwal, you get two birds with one stone.