"Odd" primary systems

[Kevinstat]

Louisiana for state elections only now: the "jungle primary" (basically a general election with party designations allowed IIRC, but with multiple candidates of the same party allowed, and a run-off between the top two candidates regardless of party if no candidate receives a majority in the "primary")

California special (non-recall) elections, at least for Congress and the Legislature: old school blanket primary except that there is no second round if a candidate gets a majority of the vote in first round (all candidates of all parties and all Independent candidates run on the same ballot, and the top polling candidates from each party and all Independent candidates advance to a runoff if no candidate receives a majority of the votes cast for all candidates; since they're special elections, you're not likely to have special primary elections where a voter could vote for a Democrat for Senator, Republican for Governor, Communist for Coroner - my main Political Science professor at UMaine Farmington loved saying that and mimicking a coroner saying in an attempted Russian accent something like "this man died due to the stresses of the capitalist system" - and I'm not sure if two special elections at the same time covering an overlapping area would use one ballot or two in that overlapping area.)

All elected offices in Nebraska except (as far as I know only) President/Vice President, U.S. Senator, U.S. Representative, Governor, Secretary of State, State Auditor, Attorney General, State Treasurer and the Public Service Commission - most notably Nebraska's nonpartisan unicameral Legislature: nonpartisan "Top-2x" primary it seems, where voters vote for x candidates, x candidates to be elected, and the top 2x candidates advance to the general election.  x is 1 for the Legislature and most offices for which this primary is used, but 2 for some positions on public power districts and 3 or 4 for some other offices, although I couldn't find any primaries for those as they only seem to be held when needed for a lot of those offices and you're not likely to find more than 6 or 8 candidates for such minor positions.

[Kevinstat]

Does each county in Nebraska have its own board of elections?  I'm sure the big cities do and the town clerks might take care of municipal elections as in Maine, but in Maine federal, state and county offices are all on the same ballot.  Thanks for the information rbt48.

[Kevinstat]

The Supreme Court has ruled that a State party may have a semi-closed primary even if this contradicts State law and the State is conducting the primary. ... But the Supreme Court has also ruled that a party may not open their primary to registrants of other parties.

Even if state law allows a party to open its primary to registrants of other parties (as Maine's does, although what constitutes the party in making the decision to send the required notice to do that isn't clear)?  Also, has the Supreme Court ruled that no voter can vote in more than one party's primary in the same primary election?  That is also allowed in Maine if voters of a certain voter class were allowed by more than two parties to vote in their primary, although I'm only aware of one party allowing non-party members to vote in their primary, and then only unenrolled voters (the Libertarians in 1992, after getting ballot access through the consent of a 1990 gubernatorial candidate - who hadn't run as a Libertarian although he described himself as such - who had received more than the required 5% of the vote to allow a new party to form without having to collect a bunch of signatures; Maine's election law was later amended to require that a party "organizing about a candidate" who got 5% or more for Governor or President use the same political designation as the candidate whose 5% allowed them to gain official ballot status, and to ban "Independent" just by itself as a designation for a political party).

In some States, unaffiliated voters may register with a party on election day, blurring the distinction between semi-closed and closed.

Maine is one of those states, and voters of other parties need to only to file an application to change enrollment (in practice a new voter registration card where you check off your new desired party and rewrite all the other required information) 15 days before a primary or municipal caucus in order to participate, although candidates for a party's nomination who were previously enrolled in another party have to have filed an application to change enrollement prior to January 1st of the election year.