With the widespread deployment of deep-learning-based speech models in security-critical applications, backdoor attacks have emerged as a serious threat: an adversary who poisons a small fraction of training data can implant a hidden trigger that controls the model's output while preserving normal behavior on clean inputs. Existing inference-time defenses are not well suited to the audio domain, as they either rely on trigger over-robustness assumptions that fail on transformation-based and semantic triggers, or depend on properties specific to image or text modalities. In this paper, we propose STEP (Stability-based Trigger Exposure Profiling), a black-box, retraining-free backdoor detector that operates under hard-label-only access. Its core idea is to exploit a characteristic dual anomaly of backdoor triggers: anomalous label stability under semantic-breaking perturbations, and anomalous label fragility under semantic-preserving perturbations. STEP profiles each test sample with two complementary perturbation branches that target these two properties respectively, scores the resulting stability features with one-class anomaly detectors trained on benign references, and fuses the two scores via unsupervised weighting. Extensive experiments across seven backdoor attacks show that STEP achieves an average AUROC of 97.92% and EER of 4.54%, substantially outperforming state-of-the-art baselines, and generalizes across model architectures, speech tasks, an open-set verification scenario, and over-the-air physical-world settings.