Recent NLP literature pays little attention to the robustness of toxicity language predictors, while these systems are most likely to be used in adversarial contexts. This paper presents a novel adversarial attack, \texttt{ToxicTrap}, introducing small word-level perturbations to fool SOTA text classifiers to predict toxic text samples as benign. \texttt{ToxicTrap} exploits greedy based search strategies to enable fast and effective generation of toxic adversarial examples. Two novel goal function designs allow \texttt{ToxicTrap} to identify weaknesses in both multiclass and multilabel toxic language detectors. Our empirical results show that SOTA toxicity text classifiers are indeed vulnerable to the proposed attacks, attaining over 98\% attack success rates in multilabel cases. We also show how a vanilla adversarial training and its improved version can help increase robustness of a toxicity detector even against unseen attacks.