The March/April 2010 issue of Biblical Archaeology Review had an article By O. Goldwasser on the creation and evolution of the Hebrew alphabet.
He writes that the earliest (around 1600 BCE) aleph-betic writings were found in the Sinai mines alongside Hieroglyphic writings. Evidently it started out as pictographic with the initial sound of the picture being the sound value of it - for example:
yud was a hand "yad" picture,
nun was a snake "nahash" picture,
mem was a water "mayim" picture,
resh was a head "rosh" picture, and so on...
The pictures gradually evolved into the Canaanite / Phoenician / Hebrew Ivri alphabet by the 10th century BCE, as evidenced by the Izbet Sartah (1200 BCE) and Tel Zayit (1100-1050 BCE) ABCeDaries, then the Gezer "calendar" (1000-950 BCE). Of Course, the Torah was given in the Early Alphabetic precursor of the Ivri script, the Ashuri characters were not in use until 500-700 years later.
The Jerusalem Talmud in Megillah 1:9 has R. Levi quoting Mar Zutra's(see DoubleAA's answer above) opinion that it was in Ivri, and the Ayin and Tet stood stood up miraculously. This makes sense only in Ivri script - Ayin is an eye/circle and Tet is a circle with an "x" in it. The Babylonian Talmud(also ref Double AA's details) has an opinion that the Mem and Samech(on Ashuri tablets) stood up, which only makes sense with the closed characters in Ashuri. To me, this situation seems perfectly logical - The Jerusalem/Israeli Rabbis of that time would still have seen plenty of evidence of the older script"s former use in the Land of Israel. On the other hand, the Babylonian Rabbis would have seen little to no former Ivri script use in Babylonia by that time, 600-1000 years after Ezra.
In Pritchard's ANET and A. Mazar's Archaeology Of The Land Of The Bible they have NO surviving Land of Israel inscriptions from before the Babylonian conquest - royal inscriptions, letters, jar handles, grave markers, you name it, that are NOT in the Ivri script.
The earliest Ashuri Hebrew/Aramaic writings are from Elephantine and Samaria from around 500-400 BCE. Dead Sea scrolls are in both, but mostly Ashuri, with only some Torah books (4Q Paleo-Leviticus, 4Q Paleo-Genesis/Exodus, etc) written entirely in Ivri.
Ashuri was evidently mostly used during the Second Temple period, except for the coins, probably because Ivri was the only proper script for JERUSALEM THE HOLY.
I hope this clears things up...By the way, I LOVE the fact that the last Israeli use of the Ivri script was when Simon Bar Kosiba took nice Roman Latin coins – hopefully a LOT of Judea Capta coins – and restamped them as Israeli coinage!
edit: added some references