Advances in 3D generation have facilitated sequential 3D model generation (a.k.a 4D generation), yet its application for animatable objects with large motion remains scarce. Our work proposes {AnimatableDreamer}, a text-to-4D generation framework capable of generating diverse categories of non-rigid objects on skeletons extracted from a monocular video. At its core, {AnimatableDreamer} is equipped with our novel optimization design dubbed Canonical Score Distillation (CSD), which lifts 2D diffusion for temporal consistent 4D generation. CSD, designed from a score gradient perspective, generates a canonical model with warp-robustness across different articulations. Notably, it also enhances the authenticity of bones and skinning by integrating inductive priors from a diffusion model. Furthermore, with multi-view distillation, CSD infers invisible regions, thereby improving the fidelity of monocular non-rigid reconstruction. Extensive experiments demonstrate the capability of our method in generating high-flexibility text-guided 3D models from the monocular video, while also showing improved reconstruction performance over existing non-rigid reconstruction methods. Project page \url{https://AnimatableDreamer.github.io/}.