Adversarial attack of structured prediction models faces various challenges such as the difficulty of perturbing discrete words, the sentence quality issue, and the sensitivity of outputs to small perturbations. In this work, we introduce SHARP, a new attack method that formulates the black-box adversarial attack as a search-based optimization problem with a specially designed objective function considering sentence fluency, meaning preservation and attacking effectiveness. Additionally, three different searching strategies are analyzed and compared, , Beam Search, Metropolis-Hastings Sampling, and Hybrid Search. We demonstrate the effectiveness of our attacking strategies on two challenging structured prediction tasks: part-of-speech (POS) tagging and dependency parsing. Through automatic and human evaluations, we show that our method performs a more potent attack compared with pioneer arts. Moreover, the generated adversarial examples can be used to successfully boost the robustness and performance of the victim model via adversarial training.