Here is the moves to the fastest stalemate I could think about:
1. e3 a5 2. Qh5 b5 3. Qxb5 c5 4. Qxc5 d5 5. Qxd5 e5 6. Qe4 f5 7. Qxf5 g5 8. Qxg5 h5 9. Qxh5+ Ke7 10. Qxh8 a4 11. Qxg8 a3 12. Qxf8+ Ke6 13. bxa3 Kd5 14. Qxd8+ Ke4 15. Qxc8 Kd5 16. Qxb8 Kc5 17. Qxa8 e4 18. Nc3 Kb6 19. Nxe4 Kc7 20. Qd5 Kb8 21. Qf7 Ka8 22. Qc7
If you could control your opponent's moves, what is the fastest stalemate you can think of?