Fine-grained software vulnerability detection is an important and challenging problem. Ideally, a detection system (or detector) not only should be able to detect whether or not a program contains vulnerabilities, but also should be able to pinpoint the type of a vulnerability in question. Existing vulnerability detection methods based on deep learning can detect the presence of vulnerabilities (i.e., addressing the binary classification or detection problem), but cannot pinpoint types of vulnerabilities (i.e., incapable of addressing multiclass classification). In this paper, we propose the first deep learning-based system for multiclass vulnerability detection, dubbed $μ$VulDeePecker. The key insight underlying $μ$VulDeePecker is the concept of code attention, which can capture information that can help pinpoint types of vulnerabilities, even when the samples are small. For this purpose, we create a dataset from scratch and use it to evaluate the effectiveness of $μ$VulDeePecker. Experimental results show that $μ$VulDeePecker is effective for multiclass vulnerability detection and that accommodating control-dependence (other than data-dependence) can lead to higher detection capabilities.