This paper presents a focused survey of error-correcting and modulation codes for spin-transfer torque magnetic random access memory (STT-MRAM), which (i) reviews representative channel models that capture write asymmetry, read disturbance, and sensing uncertainty in STT-MRAM, (ii) classifies recent error-correcting and modulation codes based on their targeted error characteristics and design strategies, and (iii) identifies future research directions involving hybrid frameworks, machine-learning-based optimization, heuristic design strategies, and quantum-inspired paradigms. The insights aim to guide the development of advanced coding techniques for enhancing the reliability of next-generation STT-MRAM systems.