The usual analysis of variance and linear regressions are not appropriate with a categorical dependent variable. One way to see the results of associations between categories and predictors is using binary logistic regression (2 outcome categories) or multinomial or nominal logistic regression (3 or more categories).
These may both be fitted in SPSS in the analyze>regression submenu. Instead of a 'F' or 't' statistic the model is assessed using a (likelihood ratio) chi-square statistic.
This statistic represents the fit between the observed outcome categories and probabilities of belonging to these categories based upon a model containing the predictor variables.
For further details with examples see Chapter 10 of Tabachnick and Fidell (2007) (available in the CBSU library).
Multigroup (also called multinomial) logistic regressions are extensions of the binary logistic regression to three or more groups. There are two approaches which differ in whether or not they assume an underlying ordering. Nominal models form the former and proportional odds models the latter. A paper illustrating a logistic model with an application to diagnosis carpal tunnel syndrome assuming an underlying ordinality is given in a pdf file located here or equivalently here and examples of both models are given in Hosmer and Lemeshow (2001). Willems and Lesaffre (2008) compare multigroup logistic models with linear discriminant analysis (see here) and conclude that logistic models are more robust to departures from normality in the responses.
Hosmer, DW and Lemeshow, S (2001) Applied logistic regression. 2nd Edition. Wiley:New York. This book is on order for CBSU library (June 2011). Note: An extended third edition of this book is due for publication in 2013.
Tabachnick, BG and Fidell, LS (2007) Using multivariate statistics fifthe edition. Pearson Education:Boston, USA.
Willems, JL and Lesaffre, E (1987) Comparison of multigroup logistic and linear discriminant ECG and VCG classification. Journal of Electrocardiology 20(2) 83-92.