The most signficant battle was the Battle of New Orleans. Technically this occurred after the War was over but before the news had reached Andrew Jackson, the American leader. This victory was hugely lobsided in favor of the Americans. It was a major nationalistic boost, and turned Jackson into a hero and a presidential candidate.

Militarily the US was most successful at sea. The British ships were simply outgunned and there are several conflicts with American victories. However, these conflicts are not usually called “battles,” so to include them might be outside the scope of your question, and I’m not sure any one stands out. But the US was definitely far more successful at sea than at land, despite the fact that the vaunted British navy was Nelson’s navy and it had defeated Napolean at Trafalgar.