10 events the sources record for this year, grouped by region.
What happened
China (3)
Chen Youliang took Taiping, proclaimed himself Han Emperor and marched on Yingtian, but was routed at Longwan by an ambush and a combined land-and-water attack. (History of Ming (Mingshi), Annals 明史 卷1 至正二十年)
Chen Youliang kills Xu Shouhui at Taiping route, proclaims himself emperor, and names the state Da Han (History of Yuan (Yuanshi), Annals 元史 卷45 順帝 1360年)
In the twelfth month, after the deaths of Guan Xiansheng and Sha Liu'er, Potouban and his deputy Zuo Li, with ten thousand from Gyeonggi, took a shortcut straight to Xijing, surrendered to Boluote'mu'er and received command. (Xu Zizhi Tongjian (Continuation of the Comprehensive Mirror) 권216 十二月 단락216.21.3)
Korean Peninsula (6)
Pansa Kim Jin, returning from Uiju, reports that the enemy has entered Seogyeong. (Goryeosa (History of Goryeo) 世家 卷39 恭愍 九年)
Our army attacks Seogyeong, and the enemy retreats to Yongkang and Hamjong. (Goryeosa (History of Goryeo) 世家 卷39 恭愍 九年)
When Hong Jeok invades Cheolhwa-po of Hwangju, the mokssa Min Hu fights, cuts off more than twenty heads and captures one. (Goryeosa (History of Goryeo) 世家 卷39 恭愍 九年)
The Japanese raid Ganghwa, kill more than three hundred people and plunder more than forty thousand seok of rice (Sim Mong-ryong cuts thirteen enemy heads and dies in battle). (Goryeosa (History of Goryeo) 世家 卷39 恭愍 九年)
May 1360: Japanese pirates raid Hoemi and Okgu in Jeolla and Pyeongtaek, Aju and Sinpyeong in Yanggwang, burning over ten counties including Yongseong. Gaegyeong is placed under martial law and Yu Tak is made commander-in-chief of Gyeonggi (Goryeosa Jeolyo (Essentials of Goryeo History) 권27 恭愍王 九年)
1360: Yi An is sent to repair the palace walls for a move of the capital to Namgyeong (Hanyang), troubling the people, but the project is halted when the divination at the royal shrine is inauspicious. Work is pushed for a new capital at Baegak, north of Imjin County (Goryeosa Jeolyo (Essentials of Goryeo History) 권27 恭愍王 九年)
Southeast Asia (1)
The king went to Tirib and Sompur. (Nagarakretagama raw 줄 1951, 필사 쪽 (118a))