In this article, the problem of recursive estimation is studied for a class of descriptor systems measured by multiple sensors with different delay rates. The delay phenomenon occurs in a random way and the delay rate for each sensor is characterised by an individual binary switching sequence obeying a conditional probability distribution. The original descriptor system is transformed into two non-descriptor stochastic subsystems with autocorrelated and cross-correlated noises by using the singular-value decomposition and the state augmentation. Applying an innovation analysis approach and the orthogonal projection theorem, recursive estimators including filter, predictor and smoother are first derived for each subsystem and the process noise. Then, based on the newly obtained recursive state estimators and process noise estimators, the recursive filter, predictor and smoother are obtained for the original descriptor system measured by multiple sensors with different delay rates. A numerical simulation example is exploited to show the effectiveness of the proposed approaches.