The best known such attempt was the so-called "business plot" of 1933, in which, it was alleged, a group of Wall Street financiers and others (supposedly including Prescott Bush, George W Bush's grandfather) sought to arrange a military coup to overthrow the Roosevelt administration and replace it with an authoritarian, pro-business regime.

The alleged plot was subject to a Congressional investigation, and it was also investigated by John L. Spivak, a well known muckraking journalist of the period, but no one ever stood trial for it and historians remain divided as to whether there was ever a serious attempt to organise a coup.

A couple of earlier AH threads give more details, and though neither offers fantastic depth, these may offer you some worthwhile reading while you wait for further contributions to this thread:

A "cocktail putsch", "a gigantic hoax", "perfect moonshine": Was the business plot a real thing? If not, why did Smedley Butler testify that it was? with u/bug-hunter

The Business Plot conspiracy in 1933. Gen. Smedley Butler claimed a plot to create a fascist veterans' organization and use it in a coup d'état seizing power over FDR, is there any new research done on affirming or discounting?, with u/InfamousBrad

Lincoln assassination? If I remember correctly it was a poorly organized attempt at a coup. The thought being that the assassination would trigger a rally of the Confederacy and motivate the scattered southern leadership to attack the financially stressed Union. The parties involved being poorly organized, emotionally motivated, and lacking real leadership.