I'd say that the most obvious and most defining purely political divide is secularism vs islamism. The sectarian divide is extremely overarching and divisive, it's also ingrained in the overall structure of the democratic process of the country. There's even an unofficial agreement that everyone follows about the distribution of the six major offices between the three main groups in the country. Shias get the prime minister role + interior ministry, Sunnis get the speaker of the parliament role + defence ministry, Kurds get the ceremonial president role + the foreign ministry. Just to present a vague idea about how ingrained sectarianism is, not a single Arab has ever been elected in the Kurdistan Region.
Voting patterns mainly reflect sectarian allegiance first, then some sort of political statement about secularism and islamism. Sunni Iraqis have mostly consistently voted in favor of secularism since 2010, with a majority of Sunnis voting for Shia political leader Ayad Allawi's secular party Wataniya in 2010, then slowly turning towards Halbusi's Sunni-led Taqaddum and PM Sudani's more secular Party recently. Shias on the other hand have a good history with islamism, unlike Sunnis who had their regions ransacked and destroyed by Sunni islamists like Al Qaeda and later ISIS. Being the comfortable majority, and willing to assert that position, Shias have mostly voted for Shia islamist candidates and parties such as Maliki's State of the Law and the various PMU-aligned parties who despite their strong anti-US positions, still maintain the most exclusionary sectarian language in the country. Kurds lean towards secular Kurdish nationalism, with the more conservative KDP dominating Erbil and Duhok, and the more leftist PUK dominating Kirkuk and Sulaymaniyah. Kurds are even more discriminatory and isolationist in their selection than Shias and Sunnis. A Shia dude can get a seat in Sunni-majority Anbar, a Sunni dude can get a seat in Basra, but an Arab will never even be allowed to run in Kurdistan, let alone get a seat.
Then comes the secondary questions such as pro-US vs pro-Iran, and economic stuff. The pro-US vs pro-Iran stuff can be 90% predicted by just looking at the secularism vs islamism position. Most secularists that won seats are naturally more pro-US, mainly because Iran is simply the main source of conservatism and islamism in the country. Being pro-Iran automatically means in most cases that a person is an islamist, with small exceptions such as the Kurdish PUK who have historically had great ties with Iran.
National politics are pretty much an extension local politics, as governorates enjoy a pretty high level of autonomy. I have never seen so much local politics penetrating the national level as in Iraq, where you have local coalitions such as Tasmim in Basra, Wasit Ajmal in Wasit and the Arab Coalition in Kirkuk actually make to the national level on an entirely local political program. The chaos basically already starts on a local level, which is why today you have a lower level of development in Shia-majority regions where the decisionmaking is stuck in the limbo of neverending bickering between the numerous similar parties, while Sunni regions are limited to 2-3 parties at most and Kurdish regions have pretty much one dominant party. If we take Sunni local politics for example, we have Anbar that is dominated by only Taqaddum and their allies, and it's rapidly advancing, while a governorate like Saladin struggles due to all the tribal beef + numerous competing parties for influence.
My overall take is that sectarian trends still mostly continue but getting a bit weaker. Secularism vs islamism continues to be the main political question for sure. The chaos of national politics is directly influenced by the chaos of local politics, as too many similar parties bicker about small issues.

Update on the Iraqi government formation:
All Sunni-led parties have formed a united Sunni front as I previously predicted. Halbusi has done it again, the fucker is an annoyingly good politician. I can't hate on him too much, his provincial government in Anbar has worked really well and him leading the Sunni vote means less influence for the more tribal and religious freaks.
The right-wing Shia parties are trying to force through Maliki again as a PM, which will fail inshallah. Maliki is working hard though, I have to give him credit, he has successfully thawed the previously dead relationship with both Barzani and Halbusi. PM Sudani sits in a weaker position than expected, most of the Shia powers seem to want a change of PM and the Sunni and Kurdish powers will go with the deal that gives them the biggest slice of the cake.