If I personally were doing it, i'd go for an allele specific PCR or something similar... only because I use PCR a lot. You could even go for a high resolution melt curve assay too, if you have access to the machine. Both assays can be done in high throughput, which will be important if you have a lot of samples.
I guess it comes down to the question being asked of your screen. Are you doing a population or diagnostic screen?
I agree with Stephen. Are you looking for new or known mutations? The other big question that nobody has addressed is of course..what type of equipment is available to you? Although HRM is probably one of the best methods for screening and detection, it requires you to have the Q-PCR instrument. Alleles specific PCR(AS-PCR) is great but it requires you to set up the assay with a positive control(i.e. a sample containing the mutation). AS-PCR does not work on unknown mutations. SSCP and DGGE are cheap alternatives for mutation screening but have the possibility of missing mutations since both methods rely on electrophoretic differences between wt sequences and mutant containing sequences.