This paper presents a comprehensive study addressing the challenging problem of acoustic howling suppression (AHS) through the fusion of Kalman filter and deep learning techniques. We introduce two integration approaches: HybridAHS, which concatenates Kalman and neural networks (NN), and NeuralKalmanAHS, where NN modules are embedded inside the Kalman filter for signal and parameter estimation. In HybridAHS, we explore two implementation methods. One is trained offline using pre-processed signals with a light training burden, while the other employs a recursive training strategy with training signals generated adaptively. The offline model serves as an initialization for recursively training the other model. With NeuralKalmanAHS, we harness the power of NN modules to refine the reference signal and improve covariance matrices estimation in the Kalman filter, resulting in enhanced feedback suppression. Our methods capitalize on the strengths of traditional and deep learning-based AHS techniques. We have explored different variants of combining Kalman filter and NN and systematically compared their howling suppression performance, providing users with versatile solutions for addressing AHS. Furthermore, by employing the proposed recursive training, we effectively mitigate the mismatch issues that plagued previous NN-based AHS methods. Extensive experimental results show the superiority of our approach over baseline techniques.