corrected AIC (AICc) assumes the model is univariate? I'm considering using the AICc instead of the AIC to select models because my sample size is not much larger than my number of parameters (n=214, K=16 - which is not enough, according to Burnham and Anderson page 66).
However, whilst they don't really say much as to the assumptions we need to make in their book, wikipedia seem to indicate some of the assumptions we need to make on the model so that the formule for AICc holds:

I don't understand what assuming that the model is univariate mean. I'm doing variable selection, across 16 variables (I plan on fixing many of them, because otherwise I'll just have overfitting) and I know that my model is not univariate. Otherwise I would just compare the 16 models with one variable.
Furthermore, I'm not sure how I can verify the other assumptions (them being: Assuming that the model [...] is linear in its parameters, and has normally-distributed residuals (conditional upon regressors)).

Here they are referring to a model in which the response variable is univariate (as opposed to a response vector) (hat tip to John Madden for pointing this out in comments).  The formula given covers simple and multiple linear regression, but not multivariate regression.
In terms of the other assumptions, you should note that the AIC and AICc can be written in terms of the underlying maximised log-likelihood function for your model.  Consequently, if you are using a regression model that is nonlinear, or which uses a non-Gaussian error distribution, you can still derive the formulae for the AIC and AICc, but you need to go through the process of looking at the log-likelihood function for your model.  An example of the derivation of the AIC (for a Gaussian regression) is in this related question.
