Which locality? I know the local politics of the town in Ohio I grew up in decently well, but I close to nothing about Madison. Perhaps that's better for my mental health.
Whichever one you know more about.
As for me, since nothing happens that's exciting in West Jordan, the suburb city in which I live in, and we have a weak-mayor, strong council form of government (city manager-style to be exact) I know literally
nothing about West Jordan politics other than I used to be represented by the State Senate President (he retired), and there was a scandal in 2000 that some old city councilperson was going on about a few months ago at the thrift store where I work. Also, they won't allow residents of the city to build solar panels on their own homes for some daft reason.
I know more about the dynamics of the state legislature though; the Democrats are obviously non-existent outside of Salt Lake City, where they rule with a (eco-friendly) iron fist. Everywhere else, the GOP easily proves that Utah is a one-party state. The only difference is in their approach to conservatism. For example, the majority of the State Senate believes that the food tax we have should be restored back to pre-2008 levels (It is currently 1.75%, Huntsman and the legislature had reduced it from 4.75%), to ensure the poor have "skin in the game" while the Utah House (or at least its speaker Becky Lockhart) wants to keep it as it is, arguing that any tax increase is bad. I think that the food tax increase will pass, because the State Senate generally has more influence than the House.
Also, we have an openly gay guy, Jim Dabakis, as State Senator in one of the Salt Lake City districts. He's also the chairman of the Utah Democratic Party. His seat is, no joke, considered by some to be the "gay seat", since it's been represented by a high-profile gay man before, and is represented in the state house by someone of the same orientation.
Oh, and West Jordan is basically either staying static or economically dying, as most everybody who lives here works in Salt Lake City, and there's no large employers here.