Here are some popular carnatic compositions on Ganesha. I had a hard time picking them as almost every composer has composed songs praising the elephant-faced deity.
1. Muthuswami Dikshithar – Vathapi Ganapathim – Hamsadhvani – Adi
2. Muthuswami Dikshithar – Siddhi Vinayakam – Shanmukhapriya (Chamaram) – Rupakam
3. Tyagaraja – Giriraja Suta – Bangala – Deshaadi
4. Swati Tirunal – Pari Paahi Ganadhipa – Saveri – Adi
5. Oothakaadu Venkatasubba Iyer – Sri Vignarajam Bhaje – Gambeeranattai – Kanta Chapu
6. Purandaradasa – Vandhi Suvudhaadhi – Nattai – Kanta Chapu
7. A collection of krithis by different composers
For a more detailed list of songs on Ganesha in different languages and composers click here
Chitra Srikrishna 