Why do Reserve Banks only move in 0.25 basis point 'leaps' It seems to me that a lot of uncertainty is created by Reserve Banks spending 5 months telegraphing "I'm gonna do it, I'm gonna raise rates", as is currently seen in the USA.

This seems to be the case where everyone knows the rate is wrong, but changing it quickly will cause shocks.

Instead of waiting 5 months and then increasing by 0.25%, why not increase by 0.05% every month for 5 months? Or smaller amounts if that is required.
The Fed normally only change rates during their regular monetary policy meetings, of which there are 8 per year (roughly six weeks between meetings). Therefore, they cannot change rates every month.

The amount 1/4% comes from interest rate quote conventions; historically they also used 1/8% (12.5 basis points). If they were to go to smaller increments, it would probably be 1/16%.

The first reason why they do not announce finer increments is that the Fed Funds rate that they target is a market interbank rate; they do not have complete control over it. The Fed publishes the "effective fed funds rate" in the H.15 report, which is their estimate of where the market is trading; it does not exactly match the target. Therefore, they cannot credibly deliver such small incremental changes to the overnight rate that matters.

The second reason is that such a small increment would probably make the Fed look silly. How do they have enough confidence in their ability to fine tune the economy so that moving the policy rate by 1/16% is now the right level?

Realistically, the current level of interest rates does not matter, so much as they expected path. At 25 bps per meeting, the total change over one year is 2%. That should be enough to match changes in economic conditions. An increment smaller than 1/8% would not be enough to keep up with conditions.

As for the current situation, it is unusual. Interest rates are very close to 0%, and it is difficult for them to go negative (the "zero lower bound"). Everyone knows that interest rates can only go one way (assuming no negative rates), so it is just a question of timing. This creates a different mood for interest rate watchers than when interest rates can go either way. Fed Reserve officials have been cautious about raising rates, since they have no leeway to cut rates if the economy rolls over. For this reason, they would not have considered any rate hikes until they were certain that rates needed to be well above 0%. (Even when the rate hikes occurred, they have been quite tentative.) Furthermore, if you want by news stories, "everyone" knew that the policy rate was "unsustainably low" every year after 2010. In other words, we need to be careful about what "everyone" knows.