If you have a structure of your protein and you want to estimate the effect of point mutations on the folding stability of your protein, there are computational methods that can do this.
FoldX is one program I've seen used for this application.
Determining the stability of an arbitrary protein sequence is more difficult. An easy, first-pass method would be to apply the
N-end rule. Here's an
online tool that uses the N-end rule and a few other simple algorithms (described in the
documentation) to predict the stability of your protein.
In practice, however, I generally don't trust the simple methods described above. What I would do is BLAST the protein sequence to see if it resembles any other proteins. If it looks to resemble a conserved domain, I'd be more likely to think that it would be stable. However, if it is full of regions predicted to be unstructured, I'd be inclined to say it is unstable. Running a secondary structure prediction is definitely a good start, especially when coupled with software that can predict ordered and disorderd regions.
In the end, I'd say there's no real right or wrong way to answer the question. Different scientists will approach the question differently, and different methods will yield different answers.