League of Nations

Both Germans and Poles claim the port city on the Polish corridor; the League of Nations creates a constitution binding both countries that makes Danzig essentially independent

based in Geneva

Swedes lose Finland and the Aland Islands, which have a high Swedish population, in 1905; the Finns and Aland Islands both declare independence after the Russiam Revolution. The British bring the issue to the League Council. So, the Council says they'll intervene but the Finns refuse, saying it's an internal matter. They agree to let the Leagye set up a neutral commission, however, which awards the islands to Finland but protects Swedish rights therein.

Final borders not yet set during Versailles negotiations; both Greek and Yugoslav troops cross the borders to protect their own boundaries.
The League of Nations sends in a commission of engineers, cartographers, and surveyors to protect pre-war boundaries and troops are withdrawn.
Highlights the theme of the use of expertise in settling disputes

Both Poland and Germany claim Upper Silesia, so the League decides to award Germany 2/3 of the land and Poland 1/3. However, the 1/3 that Poland gets contains the bulk of the resources, which preserves the balance of favor.

Memel is a valuable port city with a predominantly German population. After WWI it was under League control and granted to Lithuania. However, the French and Poles both preferred it to be an international port instead of being under the control of one country so as to ensure that it would not be closed to them, so Lithuania invaded.
The League created a commission which awarded Memel to Lithuania but mandated autonomous rights for the inhabitants, meaning the port would not be closed.

Greece invaded Bulgaria to protect boundaries (again). Bulgaria complained to the League, which told the Greeks to withdraw and the Bulgarians to stand down.

Turkey claimed Mosul, then part of Iraq, as part of their "homeland." The League sent a commission to the city asking Mosuls which country they want to be a part of. Although they would rather be independent, they choose Iraq because a) it is a British client state, which offers them protection, and b) Mosul has a large Kurdish population, who don't like Turks. The League adopts their recommendation, and although Turkey disagrees with it the higher international court says they must. Independently Iraq (really Britain) and Turkey strike a deal giving Mosul to Iraq with a time limit of 25 years.
AMBIGUITY: ultimately successful but the League was unable to solve it without traditional diplomacy, i.e. sovereign states' negotiations.

Liberia was home to the Firestone Tire Company's rubber plantations, which used forced labor (aka slavery). Liberia complained to the League of Nations, who created (yet another!) commission, who issued a report confirming the use of slavery on the plantation. The Leauge threatened to establish a trusteeship over the plantation, so the government, who had colluded with Firestone, made changes and Firestone had to leave.
NOTE: most absolute success for League because all end up happy, even Firestone

German-populated area of France that wants to be part of Germany. The League administered a fair plebiscite which showed a 90% preference to be in Germany. The decision was not disputed by any party.