Sara,
I'm assuming that you have a time series (yi,ti). If you do a least squares fit (linear or exponential) and calculate the residuals y-yi and plot them on a scatter plot vs ti any patterns either seasonal or generally periodic will be obvious because you have removed any general trend by doing the least squares fit and the subtracting the trend from the raw data. Anything other than a random pattern in the residuals indicate there is more structure to the data than just a simple trend.
Hope this helps
Jim