American forces never lost a battle in the war, so you could say militarily they had many successes. One that is commonly misconstrued as a failure is the Tet Offensive. While the attack during the Chinese Lunar New Year was a complete surprise to US forces, the Vietcong that launched the attack were routed and all but destroyed. Finally, they had come out of the jungles and were fighting US firepower head on, which was a losing proposition to be sure.
The US was also successful at modernizing South Vietnam's military with technology and training. While they could not provide the necessary motivation the South Vietnamese Army needed to fight, they were certainly given all the tools they needed.
The previous answer is correct, there were many successes. The countries nearest Indochina, most notably Thailand, the Philipines and Singapore, viewed our stand there as having stopped the spread of communism before it overpowered their countries, also. Militarily, all our large operations were successful, and most smaller ones. We developed a great deal of useful medical and technological breakthroughs due to continual developement of medical techniques, and the research and developement of sophisticated variable wing aircraft. and small arms technologies. We had a field army of quite young soldiers who were often present involuntarily, yet they performed very well overall, with kill ratios never exceeded in any other war. They did this against the VN People's Army, which from the early 1950s through the '70s was certainly the finest field infantry of the postwar period. That in itself was an American sucess, since we had trained the central cadre of that force in 1945.
Unfortunately, our failures are more glaring. The Vietnamese looked to us for protection and freedom from France at the end of WW II, and we allowed our Allies to convince us to abandon them, although the Vietnamese had helped us during the last phase of the war. We had the sense not to help the French too much, but then did not clearly see the war between the country's north and south for what it was, a civil war between the most powerful families of the northern and southern regions, which had broken out in 1776 and continued until 1975. I strongly suggest reading The Ten Thousand Day War, a good overview based on the PBS documentary series.
There are several successes that come to mind even though many still look upon this war as a negative intervention in American History. The willingness of the US to help a country that may be overcome by a Communist regime, gave a perception that we as a country would defend freedom and fight oppression among the peoples of the world. Many politicians have used this war to make wiser decisions in enlisting our military in international conflicts. On a more individual note, there are many stories of heroism where individuals have helped the refugees and oppressed in Vietnam. One in particular is a nurse who rescued an orphan an brought her to the US where she received love, education, and care from her adoptive family. This is documented at the Museum of American History in Washington DC, in the exhibit The Price of Freedom. There are also records of military successes listed in the link below.