As an important and challenging problem, multi-domain learning (MDL) typically seeks for a set of effective lightweight domain-specific adapter modules plugged into a common domain-agnostic network. Usually, existing ways of adapter plugging and structure design are handcrafted and fixed for all domains before model learning, resulting in the learning inflexibility and computational intensiveness. With this motivation, we propose to learn a data-driven adapter plugging strategy with Neural Architecture Search (NAS), which automatically determines where to plug for those adapter modules. Furthermore, we propose a NAS-adapter module for adapter structure design in a NAS-driven learning scheme, which automatically discovers effective adapter module structures for different domains. Experimental results demonstrate the effectiveness of our MDL model against existing approaches under the conditions of comparable performance.