Religious occasions
22 holy days across Hindu, Muslim, Sikh, Christian, Jain and Buddhist traditions, with the date each falls on this year and what is appropriate to write to someone who keeps it.
These are the days of observance rather than celebration: fasts, vigils, prayer, remembrance. Several of them are not "happy" occasions at all, and a cheerful greeting on the wrong one is the most common mistake people make in a mixed workplace.
Days that are not festive
Muharram is a period of mourning for Shia Muslims, marking the death of Imam Husayn at Karbala. "Happy Muharram" is wrong. Good Friday commemorates the crucifixion and is solemn; Easter Sunday two days later is the celebration. Paryushan and Samvatsari close with Jains asking forgiveness of everyone they may have wronged, which is why "Micchami Dukkadam" is the greeting rather than congratulations.
Ramadan is a month of fasting, and the celebration comes at the end with Eid al-Fitr. Wishing someone "Ramadan Mubarak" at the start is right; treating the whole month as a festival is not.
Days that are
Eid al-Fitr and Eid al-Adha, Guru Nanak Gurpurab, Christmas, Buddha Purnima, Krishna Janmashtami and Maha Shivaratri all carry warm greetings, though several involve fasting or a night vigil rather than a party.
Writing across faiths
Name the day, not the religion. "Eid Mubarak" reads better than "happy festival". If you are unsure whether a colleague observes something, a general wish addressed to them and their family is safe and rarely unwelcome.
Two practical notes: several of these dates depend on a moon sighting or a local calendar and are only confirmed a day or two ahead, and food gifts need care. Check before sending sweets to a household that is fasting, or anything with gelatine, egg or alcohol.