1. Chandra or Soma (the Moon god)
As per the chapter 'THE ORIGIN OF SOMNATH' of Shiva Purana - Part 4 (KOTI RUDRA SAMHITA)
Chandra or Soma (the Moon god) had many wives, one of whom was Rohini. Soma loved Rohini very much, which made the rest of his wives very jealous and angry. They went to their father - Daksha and complained about Soma's behaviour. Daksha went to Soma and advised him to give proper attention to all of his wives. But it did not have any effect on him and he continued giving special treatment to Rohini. When Daksha came to know about this, he cursed Soma to become weak and devoid of radiance. Then, Soma went to Prabhas area and chanted Mahamrityunjaya mantra. Lord Shiva appeared before him and asked him to demand anything he wished (boon).
Soma requested lord Shiva to liberate him from the curse given by Daksha. Hence, a clash between curse and boon (i.e Deadlock).
How that situation got resolved
Lord Shiva told Soma (the Moon god) that the words of Daksha can never became untrue. However he blessed Soma by saying that he would wane during the dark lunar phase due to the curse, but wax during the dark lunar phase due to his (Shiva's) blessings (boon).
2. Daksha
As per the "Sati Khand" of Part 2 (RUDRA SAMHITA) of Shiva Purana
After Sati jumped into the fire during Daksha Yajna. Lord Shiva gave a curse to Daksha that he must be killed and sent Veerbhadra to kill him. From the fear of Veerbhadra, Daksha prayed Lord Vishnu and got a boon that his life will be saved by Lord Vishnu. Lord Vishnu decided to fight on the side of Daksha. But Veerbhadra pulled Daksha out and severed his head.
But Lord Vishnu gave Daksha a boon that his life will be saved then how he can be killed. But Lord Shiva gave him a curse that he will be killed so how his life can be saved. Hence, a clash between curse and boon (i.e Deadlock).
How that situation got resolved
Lord Vishnu asked Lord Shiva to give Daksha's life back. Daksha was absolved by Lord Shiva and the head of a male goat was fixed on the decapitated body of Daksha and gave his life back.
Note: Narada, who was listening to the divine tales of Shiva with rapt attention, was very curious to know about the reasons why Lord Vishnu attended a yagya where Shiva was not invited and why did he fight a battle against Veerbhadra. Lord Brahma told him that all this happened due to the curse of Sage Dadhichi. The story of the curse is also given in the "Sati Khand" of Part 2 (RUDRA SAMHITA) of Shiva Purana. As most of the incidents are repetitive hence other Purana might depict the story from some other Kalpa and may not mention that Lord Vishnu fought from Daksha's side.
3. Banasura
Described in Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.63
4. King of Kashi
This story might not have scriptural basis; it may just be folklore. It is discussed here.
There may be many more instances of such Deadlocks, one I heard was King Veermani and Lord Shri Rama's battle where Lord Shiva helped Veermani in the same way how he helped Banasura and result was also similar.
This question is partially related with the below question:
Why did Hanuman fight with Lord Rama? (question's title is wrong)