Mucus is a viscoelastic aqueous fluid that participates in the protective barrier of many mammals' epithelia. In the airways, together with cilia beating, mucus rheological properties are crucial for lung mucociliary function, and, when impaired, potentially participate in the onset and progression of chronic obstructive pulmonary disease (COPD). Samples of human mucus collected are inherently contaminated and are thus poorly characterized. Human bronchial epithelium (HBE) cultures, differentiated from primary cells at an air-liquid interface, are highly reliable models to assess non-contaminated mucus. In this paper, the viscoelastic properties of HBE mucus derived from healthy subjects, patients with COPD and from smokers are measured. Hallmarks of shear-thinning and elasticity are obtained at the macroscale, whereas at the microscale mucus appears as a heterogeneous medium showing an almost Newtonian behaviour in some extended regions and an elastic behaviour close to boundaries. In addition, we developed an original method to probe mucus adhesion at the microscopic scale using optical tweezers. The measured adhesion forces and the comparison with mucus-simulants rheology as well as mucus imaging collectively support a structure composed of a network of elastic adhesive filaments with a large mesh size, embedded in a very soft gel.