You can use the Spearman Correlation if your data are measured at the ordinal, interval, or ration levels. The good news is that there are no distributional assumptions. However, if your data are measured at the nominal level meaning you cannot attribute quantity to them, you cannot use Spearman. If your data are measured at the interval or ratio levels, you will lose information and precision by using Spearman. In those cases, you should consider Pearson, but Pearson does require distributional assumptions.
A Spearman correlation is simply a Pearson correlation following a rank transformation. It doesn't require ordinal data (as noted above). If the problem is distributional you can also bootstrap CIs for a Pearson correlation (as Pearson's r doesn't assume normality except for inference). There are also other options such as various forms of robust correlation and regression.