The festival associated with Lord Shiva, often observed with fasting and night-long vigils, is:

Mahashivaratri
Navratri
Durga Puja
Ganesh Chaturthi

The correct answer is A) Mahashivaratri.

Mahashivaratri is a Hindu festival that celebrates the union of the god Shiva and his consort Parvati. It is observed on the 14th night of the dark half of the month of Phalguna (February-March) according to the lunisolar Hindu calendar. The festival is marked by fasting, vigils, and rituals dedicated to Shiva.

Navratri is a nine-day Hindu festival that celebrates the victory of good over evil. It is observed in the months of September or October according to the Gregorian calendar. The first three days of Navratri are dedicated to the goddess Durga, the next three days are dedicated to the goddess Lakshmi, and the last three days are dedicated to the goddess Saraswati. The festival is marked by prayers, fasting, and feasting.

Durga Puja is a ten-day Hindu festival that celebrates the victory of the goddess Durga over the demon Mahishasura. It is observed in the month of October or November according to the Gregorian calendar. The festival is marked by prayers, fasting, feasting, and cultural events.

Ganesh Chaturthi is a ten-day Hindu festival that celebrates the birth of the god Ganesha. It is observed in the month of August or September according to the Gregorian calendar. The festival is marked by prayers, fasting, feasting, and cultural events.