What compromises were proposed by politicians in the immediate lead up to the American Civil War, and why were they rejected or otherwise unsuccessful?

After Lincoln was elected, States started seceding. It was clear that war was a possibility.

Several peace conventions took place and representatives arrived from each state.

In summary, many Southerners were determined to secede regardless of what sort of concession was offered. They stated they only came to the peace convention to act as spies. Some Southerners were nominally pro-union and honestly tried to find a compromise. Some delegates were Northern men who were Republicans. They wanted to end the spread of slavery into new territories. They had just won the election and didn't feel the need to give it all away at the threat of war and secession. Northern Democrats were more flexible on what sorts of concessions they'd agree with.

Southerners proposed a compromise where amendments and acts would be passed that:

Effectively repealed "States Rights" and reverted back to the Missouri Compromise. What this offered was that instead of a state like Nevada potentially voting to be Free, it had to be slave because it was below a certain latitude. (The South had lost out on trying to make Kansas slave via "State Rights")
Agreement that Congress would pass no law affecting slavery except with the approval of every single slave state
Strengthening the ability of slave catchers to catch runaway slaves without a trial.

Lincoln basically said he'd agree to compromises within the existing slave states (e.g. ensure they can continue to trade slaves amongst themselves) He considered the laws on escaped slaves were already settled and already in the favour of the slave states. He did not want war and did not believe he had the right to remove slavery where it existed. But he, and the Republican delegates, refused to accept the offer that slavery be extended into New Territories. They said they had a mandate from the election to stop the spread of slavery and needed to follow the will of the voters.

The convention ended up agreeing to recommend that an amendment be created which allowed the extension of slavery into new territories. It was handed to Congress to consider. Both in the House of Representatives, Senate and by Lincoln's platform, it was dead on arrival.

The extension of slavery was the key issue which led to all the agitation. It was the focus of debate for over a decade. In the confidential letters between Lincoln and A H Stevens right prior to the war, they both discussed that topic and seemed to agree it was a political division that could not be reconciled.

What I wrote is a high level summary of the events which occurred around December 1860 and January 1861. It is covered very well the book "Abraham Lincoln: A History" Volume 3. Chapters 13 through 16.

But I'd like to step back a bit here. This was the final run-up to war after secession had already occurred. Decades of compromises and broken agreements had already occurred. The South was passionate in favor of secession and slavery. They had been threatening war for years and there was really no stopping their leaders. The leadership of the South had no intent to revert back into the Union. So, these last ditch efforts at compromise and peace were just really pipe dreams.