In this paper, we provide a series of multi-tasking benchmarks for simultaneously detecting spoofing at the segmental and utterance levels in the PartialSpoof database. First, we propose the SELCNN network, which inserts squeeze-and-excitation (SE) blocks into a light convolutional neural network (LCNN) to enhance the capacity of hidden feature selection. Then, we implement multi-task learning (MTL) frameworks with SELCNN followed by bidirectional long short-term memory (Bi-LSTM) as the basic model. We discuss MTL in PartialSpoof in terms of architecture (uni-branch/multi-branch) and training strategies (from-scratch/warm-up) step-by-step. Experiments show that the multi-task model performs relatively better than single-task models. Also, in MTL, a binary-branch architecture more adequately utilizes information from two levels than a uni-branch model. For the binary-branch architecture, fine-tuning a warm-up model works better than training from scratch. Models can handle both segment-level and utterance-level predictions simultaneously overall under a binary-branch multi-task architecture. Furthermore, the multi-task model trained by fine-tuning a segmental warm-up model performs relatively better at both levels except on the evaluation set for segmental detection. Segmental detection should be explored further.