Alternate Scenario #1: Bouyed by the success of the League of Nation's acceptance by the American people, President Woodrow Wilson does not have a stroke in the fall of 1919 while campaigning for the League across America.
In an alternate scenario, Wilson successfully negotiates a compromise with the League's leading opponents, Senators Henry Cabot Lodge and William Borah, over Article X of the Covenant, which bound the U.S. to defend a member nation who experienced aggression or attack.
Wilson's stroke probably contributed to his failure to reach a compromise with Lodge, Borah and other American opponents of the European agreements. In November 1919, the U.S. Senate refused to ratify the Treaty of Versailles or the League of Nations.
1920: League of Nations is formed, but with a different, more successful structure, and in our alternate scenario, with the support of the United States. In reality, the US Senate rejected the treaty by seven votes in March 1920; in an alternate history, the Senate accepts the treaty. The League evolves and succeeds in stopping World War II.
Alternate Scenario #2: Prohibition of the sale of alcoholic beverages, or the 18th Amendment to the Constitution, does not pass. It's actually hard to imagine a credible scenario in which it doesn't pass, given the powerful Temperance Movement throughout the U.S.
Wilson did veto the Volstead Act enforcing the national ban on the sale of alcohol, but his veto was overridden easily. Two thirds of the states enacted Prohibition; only two states rejected it. Prohibition went into effect in 1920, and was not repealed until 13 years later, after a stupendous rise in organized crime.
Alternate Scenario #3: The 19th Amendment to the Constitution, granting women the right to vote, does not pass, as it actually did narrowly pass in August 1920 when 50 of Tennessee's 99 legislators voted yes, after a nearly 40-year effort.
Alternate Scenario #4: President Warren G. Harding does not die in office at age 57 in 1923, but endures multiple scandal revelations in 1923 and 1924. Facing re-election and a Progressive Party opponent, Bob LaFollette, as well as a Democratic opponent, John Davis, he still manages to win re-election because the economy is booming, proving that voters care far more about the economy than a politician's honesty. (Actually, President Calvin Coolidge does face these opponents and wins handily.)
Alternative Scenario #5:
1923: Turkey does not win its war for independence and Greece takes Constantinople (Istanbul) and a chunk of what is now Western Turkey. Greek Christians in Turkey are not forced out, and Muslims in Greece are not forced into Turkey. (See population exchanges.) However, would this have led to greater killing or even genocide of Greek Christians in Turkey and Muslim Ottoman Turks in Greece?
Alternative Scenario #6
1923: Fascist Benito Mussolini does not come to power in Italy, or is prevented from defying the constitution in 1925 and setting up a one-party dictatorship. In 1940, Germany is deprived of an alliance with fascist Italy, and Nazi Germany falls far sooner than 1945.
Alternative Scenario #7:
1924: The leader of the communists, and first head of the Soviet Republic in Russia, Vladimir Lenin, does not die after suffering from years of declining health; or his preference that Trotsky succeed him is fulfilled; or his rejection of Stalin as successor is understood and agreed to by the Politburo.
Alternative Scenario #8
1927: Chinese Civil War does not start. Any chance for peaceful transition?
Alternative Scenario #9
1927: Charles Lindburgh's plane crashes, and the evolution of air flight halts dramatically.
Other plausible alternate scenarios from the 1920s?
I can't think of any. Herbert Hoover won the 1928 presidential election in a landslide against Al Smith. It's impossible to imagine the Democrats winning that year, unless you change the timeline for the Great Depression, perhaps if the stock market crashed in early to mid-1928 rather than 1929. But writing a plausible scenario requires an intimate knowledge of economics and a willingness of the US government to intervene in the economy. It was necessary for the nation and the world to endure years of economic depression before policy-makers would consider ending "laissez-faire" economics.
Not until 1933 did economist John Maynard Keynes start a revolution in thinking about economics by writing the treatise that gave Franklin Roosevelt the intellectual justification for stimulating the economy and reducing unemployment through deficit spending.