I just wrote a research paper on the Iran-Iraq War. If sources are requested I could provide them.
One of the reasons Saddam attacked Iran to begin with is that Iran was calling for Iraqi Shiites to overthrow the Iraqi government. Iran even went as far as training Shiite guerilla fighters and sent them across the border into Iraq. For this and other reasons, Saddam invaded Iran. At first he did have difficulty getting his army, which was 80% Shiite, to fight hard. But after 1982 when Iran invaded Iraq the Iraqi army began fighting much harder. This happened for a couple of reasons. First, at this point in the war Saddam recognized that he needed to shore up support if he was going to survive this war, so he reconfigured the government and allowed several Shiites into positions of power. Second, he reformed the army and began paying his troops better. These two things, combined with the fact that they were now defending their own land, motivated the Iraqi Shiites to fight harder.
Also, even before the war Saddam recognized that he had to improve his position in southern Iraq, so he spent considerable amounts of the countries income in modernizing that part of the country.
In short, before and during the war Saddam was aware of the threat of a revolution in the south, and he took measures to more or less buy off the Shiite community there.
(Edit) Since sources have been requested:
Hiro, Dilip. The Longest War: The Iran-Iraq Military Conflict
Pelletiere, Stephen C. The Iran-Iraq War: Chaos in a Vacuum
Takeyh, Ray. "The Iran-Iraq War: A Reassessment."
I did my undergrad thesis on the Iran-Iraq war. From what I remember, the Ba'athist government had been rather effective in suppressing Shia dissent in the late 1970s by encouraging economic development in the south, bribing Shia leaders, and of course some violence/arrest of radicals.
Ethnic nationalism may have also trumped religious ties. Saddam certainly tried to portray the war as an ethnic one, in which Islamic Arab civilization was battling hoards of Zoroastrians from the east. So the more nationalistic Shia may have bought into this narrative.
Then of course there were rivalries ( I'm not sure what they stem from exactly) between the Iranian and Iraqi clergy. Najafi/Karbali clerics probably didn't like the idea of their centers of power being "liberated" by upstarts from Qom/Mashad.
Short answer: He wasn't. The regime was simply very effective in suppressing these uprisings.
##During the war
A non-negligible number of Shia Iraqis fought on the side of the Iranian government during the Iran-Iraq War. This was onset by a few major events, not least the establishment of the Supreme Council for the Islamic Revolution in Iraq (SCIRI), which was formed in Iran by an exiled Shia Iraqi cleric named Mohammed Baqir al-Hakim. This organization served as a very important proxy for Iran in Shia-dominated areas in southern Iraq, and acted as a sort of central organization for the efforts of Shia Iraqi Islamists to combat the Ba'athist Saddam regime during the war.
The establishment of the SCIRI was preceded closely by an attempt from Shia gunmen in the predominantly Shi'ite town of Dujail to assassinate Saddam Hussein during his tour of the town in 1982. In response to this assassination attempt, Saddam ordered the detention and torture of nearly 1,000 of the town's citizens. Many of these detainees were exiled, while some were released, and nearly 150 were sent to trial, charged with treason, and executed. This crackdown was simply a continuation of the tone Saddam's regime had historically taken towards dissent and uprising, which was to respond quickly and forcefully. This response served as an important demoralizer for the nation's Shia populace in any attempts they made to fight against the Iraqi government or express support for Iran in the conflict.
However, the Iraqi government did attempt to coax many of its Shia citizens into support for Iraq over Iran through more political means. The government focused economic resources towards the southern, Shia-dominated regions of the country, and ran targeted propaganda emphasizing the importance of supporting Iraq in the conflict. Saddam even offered positions in power to Shia leaders, and attempted to retain many of the Iraqi Army's forces through general overhead policy changes. This tactic of responding forcefully to uprisings while politically herding Shia Islamists was not unheard of during the Saddam regime, and the government used it effectively during the Iran-Iraq war throughout the 1980s in order to remove the outright possibility of facing a war on two fronts -- one with Iran, and one with its populace.
##Post-war
As you may recall, the Iran-Iraq War ended in a stalemate with hundreds of thousands of casualties -- both civilian and military -- on both sides of the conflict. This stalemate was very important in redefining the landscape of Iraq post-war, and allowing for more permeability in regards to a Shia populace yearning for regime change.
The relative state of economic and social weakness faced by the Iraqi government in 1991, as a result of the Iran-Iraq War and its '90-'91 invasion of Kuwait, led to a wave of uprisings across the nation spearheaded predominantly by the country's Shia populace but also by Kurdish nationalists. Over a period of about a month, many citizens revolted against the Iraqi government, killing around 5,000 officers and soldiers of the Republican Guard and Iraqi Army and seizing several important cities, including Karbala and Kirkuk. However, the tide quickly turned as Saddam organized a central opposition and responded in force to the uprisings, killing between 100,000 and 200,000 civilians and retaking nearly every single city which had initially been lost to rebel forces.
It was in the aftermath of the 1991 uprisings that Saddam reinforced his policy of retribution for physical opposition. The Marsh Arabs, a group of Shia Islamist Arabs residing in the Tigris-Euphrates wetlands who were instrumental in providing support for rebels during the uprisings, were systematically displaced after the Iraqi government drained the wetlands as punishment for their role in the uprisings.
After 1991, there were no more significant Shia uprisings in Iraq until the U.S.-led invasion in 2003, barring a minor and relatively quickly dispersed uprising in 1999 following the death of Shia cleric Mohammad Mohammad Sadeq al-Sadr.
##Afterthoughts
As I mentioned in the short answer at the top of this comment, a number of fairly significant uprisings among Iraq's Shia population did occur throughout the Iran-Iraq War and afterwards throughout the 1990s. However, the Iraqi government's policy of responding quickly and forcefully to dissent played a very large role in actively suppressing these uprisings and discouraging any future uprisings, as well as demoralizing the Shia populace in their support for Iran during the war.
Thankyou for the fantastic answers /u/Hey_Arnoldo, /u/probiotic_asgard, /u/BipolarBear0 and any others I have missed.
This goes a long way to improving my understanding of the current situation in Iraq