When was the U.S. Constitution signed?
It was signed on by the delegates to the Constitutional Convention, meeting in Philadelphia, on September 17, 1787. Contrast this with the signing of the Articles of Confederation, the prior constitution of the United States. It was first signed on July 9, 1778, but due to a number of objections by the states (particularly over the status of the western territories), signing was not completed until March 1, 1781.
However that is not the date that the United States was actually established; in other words, the Constitution did not take effect on that date. Technically, and according to its own stipulation, it took effect when it was signed by the ninth state to ratify it: New Hampshire. That happened June 21, 1788. The date mentioned above, September 17, 1787, is when the Constitutional Convention in Philadelphia agreed on the final draft of the Constitution and sent it to Congress with the recommendation that it be sent to the states for ratification.