The key assumption in the MNL is that the errors are independently and identically distributed with a Gumbel extreme value distribution. The problem with testing this assumption is that it is made a priori. In standard regression you fit the least-squares curve, and measure the residual error. In a logit model, you assume that the error is already in the measurement of the point, and compute a likelihood function from that assumption.

As far as assumptions on the model itself, Train describes three:

1. Systematic, and non-random, taste variation.
2. Proportional substitution among alternatives (a consequence of the IIA property).
3. No serial correlation in the error term (panel data).

The first assumption you mostly just have to defend in the context of your problem. The third is largely the same, because the error terms are purely random.

The second is testable to a certain extent, however. If you specify a nested logit model, and it turns out that the inter-nest substitution pattern is entirely flexible ($\lambda = 1$) then you could have used the MNL model, and the IIA assumption is valid. But remember that the log-likelihood function for the nested logit model has local maxima, so you should make sure that you get $\lambda =1$ consistently.

As far as doing any of this in SPSS, I can't help you other than suggest you use the mlogit package in R instead. Sorry.

The key assumption in the MNL is that the errors are independently and identically distributed with a Gumbel extreme value distribution. The problem with testing this assumption is that it is made a priori. In standard regression you fit the least-squares curve, and measure the residual error. In a logit model, you assume that the error is already in the measurement of the point, and compute a likelihood function from that assumption.

An important assumption is that the sample be exogenous. If it is choice-based, there are corrections that need to be employed.

As far as assumptions on the model itself, Train describes three:

1. Systematic, and non-random, taste variation.
2. Proportional substitution among alternatives (a consequence of the IIA property).
3. No serial correlation in the error term (panel data).

The first assumption you mostly just have to defend in the context of your problem. The third is largely the same, because the error terms are purely random.

The second is testable to a certain extent, however. If you specify a nested logit model, and it turns out that the inter-nest substitution pattern is entirely flexible ($\lambda = 1$) then you could have used the MNL model, and the IIA assumption is valid. But remember that the log-likelihood function for the nested logit model has local maxima, so you should make sure that you get $\lambda =1$ consistently.

As far as doing any of this in SPSS, I can't help you other than suggest you use the mlogit package in R instead. Sorry.