Saul is rejected twice, and both occasions have troubled readers because the offences look small beside the sentence.
At Gilgal he is waiting for Samuel to come and offer sacrifice before an engagement. Samuel is late by a day, the army is deserting, and Saul makes the offering himself. Samuel arrives as he finishes and tells him his kingdom shall not continue.
The second is the campaign against Amalek, where the instruction is total destruction, people and livestock together. Saul defeats them, spares the king Agag, and keeps the best of the sheep and cattle, saying they were for sacrifice. Samuel answers with the line that has outlived the episode: to obey is better than sacrifice, and to hearken than the fat of rams. As he turns to go Saul catches his robe and it tears, and Samuel makes the tear the sign that the kingdom has been torn from him. Then Samuel kills Agag himself.
Neither passage lets the modern reader off. The first sentence falls on a man doing something a king might reasonably do; the second falls on a man who declined to complete an order for the killing of a whole population, and the text treats the incompleteness as the sin.
The books do not soften this, and they carry more than one account of how Saul became king in the first place: anointed privately, chosen by lot at Mizpah, and acclaimed after a victory. The founding of the monarchy is preserved as contested material, and its first king is judged by a standard the narrative never explains.