Segmentation is an important task in a wide range of computer vision applications, including medical image analysis. Recent years have seen an increase in the complexity of medical image segmentation approaches based on sophisticated convolutional neural network architectures. This progress has led to incremental enhancements in performance on widely recognised benchmark datasets. However, most of the existing approaches are computationally demanding, which limits their practical applicability. This paper presents an expand-squeeze dual multiscale residual network (ESDMR-Net), which is a fully convolutional network that is particularly well-suited for resource-constrained computing hardware such as mobile devices. ESDMR-Net focuses on extracting multiscale features, enabling the learning of contextual dependencies among semantically distinct features. The ESDMR-Net architecture allows dual-stream information flow within encoder-decoder pairs. The expansion operation (depthwise separable convolution) makes all of the rich features with multiscale information available to the squeeze operation (bottleneck layer), which then extracts the necessary information for the segmentation task. The Expand-Squeeze (ES) block helps the network pay more attention to under-represented classes, which contributes to improved segmentation accuracy. To enhance the flow of information across multiple resolutions or scales, we integrated dual multiscale residual (DMR) blocks into the skip connection. This integration enables the decoder to access features from various levels of abstraction, ultimately resulting in more comprehensive feature representations. We present experiments on seven datasets from five distinct examples of applications. Our model achieved the best results despite having significantly fewer trainable parameters, with a reduction of two or even three orders of magnitude.