Objective: Detection of Electrocardiogram (ECG) characteristic points can provide critical diagnostic information about heart diseases. We proposed a novel feature extraction and machine learning scheme for automatic detection of ECG characteristic points. Methods: A new feature, termed as randomly selected wavelet transform (RSWT) feature, was devised to represent ECG characteristic points. A random forest classifier was adapted to infer the characteristic points position with high sensitivity and precision. Results: Compared with other state-of-the-art algorithms' testing results on QT database, our detection results of RSWT scheme showed comparable performance (similar sensitivity, precision, and detection error for each characteristic point). RSWT testing on MIT-BIH database also demonstrated promising cross-database performance. Conclusion: A novel RSWT feature and a new detection scheme was fabricated for ECG characteristic points. The RSWT demonstrated a robust and trustworthy feature for representing ECG morphologies. Significance: With the effectiveness of the proposed RSWT feature we presented a novel machine learning based scheme to automatically detect all types of ECG characteristic points at a time. Furthermore, it showed that our algorithm achieved better performance than other reported machine learning based methods.