Undoubtedly you'll get almost as many different answers as respondents! I tend to agree with Felipe, since the high security straws made in France are the safest in terms of breakage, integrity and protection from (theoretical) cross-contamination by viruses. On the other hand, nothing beats internally threaded cryo vials for convenience. Their main downfall is that they are designed to be used in nitrogen vapors, and may allow liquid nitrogen to enter the vial if stored in liquid phase. That hasn't stopped many people from doing that. Also, they are available in several appropriate volumes (e.g., 0.5, 1.0, 1.5, 2.0 mL), whereas the French straws are now only available in 0.5 mL or smaller - not as useful for straight semen freezing for humans or most domestic mammals. I would recommend using vials from a well-established company rather than re-branded ones. And, always seal the tops correctly: not too tight, not too loose. The gasket should not be squeezed out beyond the vial's diameter.
I agree with charles and Felipe. Straw or vials depending what type of cryo tank you will be storing your samples. The taylor-wharton k- series can store both vials and straws.
For many species the best cryovial is a glass vials (2 or 8 ml) used with a directional freezer, it has giving us consistently better results than straws for the species we work with (mammals and birds).
Many sperm banks use Nunc vials for human semen, although they may store in vapor. Look for vials from this or other companies which have a cap that does not allow contents to enter it.
I agree with Charles and Felipe. I could only add that it is OK if You would like to freeze human sperm. If You would like to preserve semen of other species, many things depends on species-dependent 'freezability'. Generally less volume vials with higher surface/volume ratio are better if freezability is poorer
In avian semen, depending on freezing protocols (conventional freezing on nitrogen vapors or pellets produced by dropping semen in liquid nitrogen) we use both french straws and plastic cryovials with cap, respectively.
TPP brand cryo tubes made in Switzerland are best quality cryotubes (1ml to 4ml). These tubes comes with almost zero phthalates leachables. Hence, bio-material are safe to revive even after decades! storage