This is the dataset accompanying the paper "Parameter identifiability and model selection for partial differential equation models of cell invasion" (https://arxiv.org/abs/2309.01476). It consists of a series of images taken of a barrier assay experiment to study tissue expansion of MDCK cells, along with cell density data in MATLAB format. File structure: the contents of the four zip files should be combined (they were split into four files for practical reasons regarding file size). The data corresponds to eight experiments, four with circular initial conditions, and four with triangular initial conditions, the associated data are located in 04-05-22 exp1/Circle and 04-05-22 exp1/Triangle respectively, each labeled "xy<n>", where <n> from 1 to 8 is an identifier for the experiment. The images under the xy<n>_Phase folders are the raw images taken of the experiment, those under the xy<n>_mask folder are processed images indicating the extend of the spread of the cell population. The DensityCellcyleFraction folder contain data files in MATLAB format. The most relevant is the "density" variable, which is a rank-3 tensor of size 150x150x77 such that density(i,j,k) corresponds to the cell density at location (x_i,y_j) and time t_k. The process for calculating the cell density is described in the paper. Alternatively, the density data is also provided in csv format. In the csv_data folder, xy<n>/t<k>.csv encodes a matrix representing cell density for experiment <n> at time t_k. The code for processing and analysing these data are provided at https://github.com/liuyue002/woundhealing . Abstract of the paper: When employing a mechanistic model to study biological systems, practical parameter identifiability is important for making predictions in a wide range of scenarios, as well as for understanding the mechanisms driving the system behaviour. We argue that parameter identifiability should be considered alongside goodness-of-fit and model complexity as criteria for model selection. To demonstrate, we use a profile likelihood approach to investigate parameter identifiability for four extensions of the Fisher--KPP model, given experimental data from a cell invasion assay. We show that more complicated models tend to be less identifiable, with parameter estimates being more sensitive to subtle differences in experimental procedures, and require more data to be practically identifiable. The results from identifiability analysis can inform model selection, as well as data collection and experimental design.