Which of the Islamic Empires (Ottomans, Mughals, Safavids) were the most influential and why?

The correct answer is - Ottomans.

The Ottomans were the most influential Islamic Empire of the three that are suggested. The reason for that is that the Ottomans had the largest territory, spread out on three continents, and managed to preserve and strengthen the Islam in a period when the spreading of this religion was mostly stopped.

The Ottomans also managed to capture two of the three most important cities for the Christians, Jerusalem and Constantinople, which was a big victory for the Islam.

They managed to spread the Islam in places where it didn't managed to spread previously, like the Caucasus and the Balkans.