Despite being spoken by millions of people, Tigrinya remains severely underrepresented in Natural Language Processing (NLP) research. This work presents a comprehensive survey of NLP research for Tigrinya, analyzing over 50 studies from 2011 to 2025. We systematically review the current state of computational resources, models, and applications across fifteen downstream tasks, including morphological processing, part-of-speech tagging, named entity recognition, machine translation, question-answering, speech recognition, and synthesis. Our analysis reveals a clear trajectory from foundational, rule-based systems to modern neural architectures, with progress consistently driven by milestones in resource creation. We identify key challenges rooted in Tigrinya's morphological properties and resource scarcity, and highlight promising research directions, including morphology-aware modeling, cross-lingual transfer, and community-centered resource development. This work serves both as a reference for researchers and as a roadmap for advancing Tigrinya NLP. An anthology of surveyed studies and resources is publicly available.