What statistical method can be applied in my case?
There are a number of methods for this. Here's a list:


You can build a Regression model and observe the p-values of the coefficients of each variable.
Pearson Correlation
Spearman Correlation
Kendall Correlation
Mutual Information
RReliefF algorithm
Decision trees
Principal Component Analysis (which you have tried)


etc. 

You can search for other methods with these keywords:

feature selection, variable importance, variable ranking, parameter selection

They are all almost the same, just different terminology in different fields.

However, the issue that you MUST pay attention here is "10" observations are almost nothing. It will be extremely difficult to trust any outcome, regardless of the method you use.