Instruction-following is a critical capability of Large Language Models (LLMs). While existing works primarily focus on assessing how well LLMs adhere to user instructions, they often overlook scenarios where instructions contain conflicting constraints—a common occurrence in complex prompts. The behavior of LLMs under such conditions remains under-explored. To bridge this gap, we introduce ConInstruct, a benchmark specifically designed to assess LLMs' ability to detect and resolve conflicts within user instructions. Using this dataset, we evaluate LLMs' conflict detection performance and analyze their conflict resolution behavior. Our experiments reveal two key findings: (1) Proprietary LLMs exhibit strong conflict detection capabilities, with Claude-3.5-Sonnet and GPT-4o achieving average F1-scores of 86.6\% and 84.9\%, ranking first and third, respectively. (2) Despite their strong conflict detection abilities, LLMs rarely explicitly notify users about the conflicts or request clarification when faced with conflicting constraints. These results underscore a critical shortcoming in current LLMs and highlight an important area for future improvement when designing instruction-following LLMs.