It's my understanding that the famous "sink swirl" example doesn't work simply because the Coriolis effect is too weak at that scale: it's insignificant compared to the motion of the water from residual momentum from when it entered the sink, and from vibration, air motion, and so on.
For long-range artillery, the Coriolis effect can become noticeable: it causes the projectile to deviate from the path that you would otherwise expect it to follow. The most famous example is probably the Paris gun. This probably doesn't constitute a practical experiment for most people, though.
To address the "or otherwise" part of your question, there are definitely simpler ways to determine your hemisphere: if the sun is to the south in the middle of the day, you're in the northern hemisphere, and vice versa. This assumes that you know where north is, but you can determine this by observing the stars. (This, of course, requires a clear and visible sky.)
If you have a way of measuring the vertical component of the Earth's magnetic field, you can also use the magnetic dip to determine your hemisphere. This won't work well if you're less than about 15° from the equator, since the magnetic equator doesn't correspond exactly to the geographical equator.