Has anyone any idea how one could distinguish time series according to certain properties? The only time series properties I know are stationarity/nonstationarity and homoskedasticity/heteroskedasticity. But are there any other possibilities to distinguish time series?
A very widely used time-series identification scheme is the Box-Jenkins approach. See here. This involves establishing stationarity and decomposing seasonality, then fitting Autoregressive (Integrated) Moving Average (ARIMA) models to the resulting series. The R function ARIMA() in the stats package will do this and chooses the appropriate model and estimates parameters according to AIC.
A point of clarification in your question, however: The concept of stationarity includes the variance (i.e. homoskedasticity vs. heteroskedasticity). A stationary series is, by definition, homoskedastic. See here.
I think I have found what I have been looking for in the paper of Rob Hyndman:
"...there are nine classical and advanced statistical features describing a time series’ global characteristics. They are: trend, seasonality, periodicity, serial correlation, skewness, kurtosis, non-linearity, self-similarity, and chaos." (Wang/Smith/Hyndman (2006),Characteristic-Based Clustering for Time Series Data)