Most of the references above are regarding the construction of the breastpiece or how it was placed in relation to the high priest's outfit. The highlighted references above may yield some other facts, so let's take a look at them more closely:
Exodus 28:15 "Fashion a breastpiece for making decisions -- the work of a skilled craftsman. Make it like the ephod: of gold, and of blue, purple and scarlet yarn, and of finely twisted linen."
[As we already noted, this breastpiece was somehow to be used in making decisions for the nation.]
Exodus 28:28-30 "The rings of the breastpiece are to be tied to the rings of the ephod with blue cord, connecting it to the waistband, so that the breastpiece will not swing out from the ephod. Whenever Aaron enters the Holy Place, he will bear the names of the sons of Israel over his heart on the breastpiece of decision as a continuing memorial before the LORD. Also, put the Urim and the Thummim in the breastpiece, so they may be over Aaron's heart whenever he enters the presence of the LORD. Thus Aaron will always bear the means of making decisions for the Israelites over his heart before the LORD."
[There was great care taken to fasten the breastpiece securely to the ephod so that it was not left swinging about in an undignified manner. Aaron bore the names of the tribes of Israel over his heart. The Urim and Thummim were put into the breastpiece, so that Aaron would always have the means of making decisions for the Israelites.]
The NIV Study Bible comments on this verse noted that "...the Hebrew for the phrase 'Urim and Thummim' probably means 'the curses and perfections'...Urim begins with the first letter of the Hebrew alphabet and Thummim begins with the last letter of the Hebrew alphabet. They were sacred lots and were often used in times of crisis to determine the will of God (see Numbers 27:21). It has been suggested that if Urim (curses) dominated when the lots were cast the answer was "no," but if Thummim (perfections) dominated it was "yes." In any event, their "every decision" was "from the LORD" (Proverbs 16:33)."
I must confess that I find this method of decision-making a bit odd. It seems that it was only used in a crisis situation, though, so perhaps it was done this way so that the people would not have to worry if the decisions being made were just coming from out of the high priest's own mind and not from God. Or perhaps it underlined the fact that they were relying upon the LORD for direction in the crisis.
Leviticus 8:8 "He placed the breastpiece on him and put the Urim and Thummim in the breastpiece."
Here are some verses where the the Urim and Thummim, or just the Urim, are mentioned. There are not a lot of such verses, but perhaps they can shed some light upon how they were used or in what kinds of circumstances they were employed:
I Samuel 28:6
Some verses where King David called for "the ephod," [which implied the accompanying articles of the high priest's garment, including the breastpiece, which held the Urim and Thummim] are as follows:
I Samuel 23:9-12
I Samuel 30:7-8
That is all that I know at this time regarding the Urim and Thummim.