Firstly, some years ago I wrote a Tutorial on Tides that ought to answer some of your questions.

When the Moon is full or new, it is in the same line as the Earth and Sun, i.e. syzygy. At the quarters, it is at quadrature.

There is a small tide in Earth's crust. This is not due, as was falsely stated elsewhere in this thread (but now, thankfully, deleted), to the Earth having a molten core, but due to the fact that the crust is slightly flexible (see note 2 in the tutorial). This crustal tide gives rise to the notion that the movement in the crust at lunar syzygy is sufficiently large to trigger earthquakes. There are two main problems with this:
Observational. Statistical analysis of earthquake frequency show no correlation to lunar syzygy. Despite what was claimed elsewhere on this MB, the links posted with that claim do not provide a statistical analysis that shows this correlation.
Mechanism. In order to trigger an earthquake, the trigger would need to apply a differential force across the tectonic plate boundary. Crustal tides do not apply a differential force: the tidal force is the same on both sides of the boundary.

If it helps, try visualising what happens. We have an approximate sphere of just under 4000 miles in radius. The difference in crustal tide between lunar syzygy and quadrature is of the order of an inch, or 0.00000004% of the radius. This is equivalent to taking a 30cm (12 inch) ruler and bending it so that it is deformed by less than a millionth of a centimetre (or inch), then increasing the bending by another less than a millionth of a centimetre (so that the total deformation is still less than a millionth of a centimetre, and expecting to notice a difference. You could make the ruler out of meringue, and that would still be insufficient to trigger a fracture, even if a fracture line already existed!

Perhaps now you understand why those who have been touting a correlation between lunar syzygy and earthquakes have not provided any supporting mathematics, either for the statistics or for a mechanism.

Note: There is a strongly significant correlation between vulcanism and lunar syzygy, but vulcanism is not the same thing as earthquakes.

However, there is a possible relationship between ocean tides and submarine earthquakes. I am certainly no expert on this and you may be better advised to get your information from someone who is better informed than I. As I understand it, it is only a suspected relationship and it is not properly understood. Again, as I understand it, the proposed mechanism relies on one plate being largely under ocean and the other being largely under land. Ocean tides can therefore create a differential loading on the plate boundary, thus triggering an earthquake. (If my understanding of any of this is wrong, I would welcome correction by someone who is more competent than I to discuss it.)

On a practical level, the (ocean) tidal hypothesis cannot be used to predict earthquakes.

If one looks at earthquake data, we note that they are distributed throughout the synodic lunar month.

I hope that helps.






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Stephen