Synaptic delay plasticity has been used to design supervised learning algorithms for spiking neural networks (SNNs). Considering both synaptic connection strength plasticity and synaptic delay plasticity in biological nervous systems, we propose an online synaptic weight-delay supervised learning algorithm based on spike train kernels for feedforward SNNs in this paper. The proposed algorithm uses the kernel function representation of spike trains to construct a real-time error function at the spike train level, and derives online learning rules for synaptic weights and delays by combining the gradient descent rule. The proposed algorithm enables online learning of spike trains, and can simultaneously adjust synaptic weights and delays between neurons during supervised learning. The learning performance of the proposed algorithm is verified by spike train learning tasks and nonlinear pattern recognition tasks on UCI datasets. The spike train learning results show that the introduction of dynamic learnable synaptic delays can effectively improve the learning performance of feedforward SNNs. The results of UCI dataset classification show that the proposed algorithm has certain advantages in solving complex spatio-temporal pattern recognition problems compared to some common supervised learning algorithms for SNNs.