Is zero inflation desirable? Is zero inflation really desirable?

To be more precise: Does inflation in real life have benefits that in some situations outweigh its social cost? E.g.: it works as a disincentive against holding money. The treasury derives income from printing new bills, which usually inflates prices.
The optimal level of inflation is very debated with unclear answers. There are many reasons, and a great answer would be very long. It should also distinguish between expected inflation and surprises.

I'm not going to do any of this, but giving you three reasons for a desirable positive level of inflation. This list is of course incomplete, also there are many reasons against too high inflation.

Downward-rigidity of wages

It is really unclear to economists why it is happening, but nominal wages seem to be downward rigid. It appears to be a behavioral thing (and might not be true at all, see Barro (1977)), but it appears that in crises, once presented with the choice of more firings or reduction of wages, most firms/workers decide to not cut wages but rather respond to the slump with increased separations.

To the extent that we believe this lack of reduction in wages is suboptimal, the central bank can enforce a reduction in real wages through increased inflation rates, and thus preventing separations to some extent.

Redistribution

This argument is based on Keynesian theory. Keynes claimed that the marginal propensity to consume out of income is smaller for rich households (and indeed, we do find this in the data  to some extent). Unexpected inflation is similar to redistribution from debtors to borrowers, as long as debt contracts are not indexed to inflation. 

To the extent that poor people consume more out of this unexpected wealth shock and than rich people decrease their consumption, this redistribution will lead to an increase in aggregate consumption.

Room to cutback interest rates

This argument was most prominently brought forward by Krugman around 2008. In recessions, you want to be able to decrease nominal interest rates in order to punish households and firms for "holding cash" and incentivize them to spend it instead. If you start with low (say, 2-3%) nominal interest rates during normal times, you don't have a lot of space to cut back nominal interest rates during the crises.

If, instead, you would have higher nominal interest rates (Krugman argued for around 8%) and relatedly higher inflation during normal times, you could easier cut back nominal interest rates during busts and stimulate the economy.