As "significance" is a fundamentally subjective or relative characteristic, this question can only be answered with suggestions as to what have been considered major challenges; no one can be considered in absolute terms "the most significant."
The post-World War II period is sometimes know as the period of the "Cold War" in which the US and the USSR were competing for global influence and power. For many, this was a significant challenge for both nations.
Another major challenge was rebuilding the economy after the Great Depression; although the economy recovered in the war due to a massive influx of government spending, shifting the economy to a stable peacetime footing was still a major challenge.
Another challenge was that the US had finally become a leading power on the world stage and needed to learn how to adjust to its new position.
Finally, this was the period of major civil rights movements, including racial desegregation, women's rights, gay rights, etc. Learning how to create a fair multicultural society without gross inequalities and injustices is an ongoing challenge.