Compilers are essential for the performance and correct execution of software and hold universal relevance across various scientific disciplines. Despite this, there is a notable lack of tools for testing and evaluating them, especially within the adaptable Multi-Level Intermediate Representation (MLIR) context. This paper addresses the need for a tool that can accommodate MLIR's extensibility, a feature not provided by previous methods such as Csmith. Here we introduce MLIR-Smith, a novel random program generator specifically designed to test and evaluate MLIR-based compiler optimizations. We demonstrate the utility of MLIR-Smith by conducting differential testing on MLIR, LLVM, DaCe, and DCIR, which led to the discovery of multiple bugs in these compiler pipelines. The introduction of MLIR-Smith not only fills a void in the realm of compiler testing but also emphasizes the importance of comprehensive testing within these systems. By providing a tool that can generate random MLIR programs, this paper enhances our ability to evaluate and improve compilers and paves the way for future tools, potentially shaping the wider landscape of software testing and quality assurance.