To decide whether zero inflation is desirable or not we should first qualify the question. Inflation is not a universal constant neither is the economic universe immutable so as to admit some particular fixed inflation level. Furthermore, inflation has different effects on different groups of agents; these depend on the composition of their wealth (debt or equity) and the nature of their incomes. Additionally, a second order issue might arise with the kind of time preferences that different agents employ when discounting the future.
Since late 2007-early 2008, the world economy has been perturbed severely by what has come to be known as the great recession. One major side-effect of this has been the systematic difficulty with which a number of Central Banks struggled to achieve the objectives of their mandates ie financial stability with subdued inflation.
In the past, the 'problem' with inflation had been its tendency to spiral uncontrollably upward. In this context, most recall instances before the Great Depression in the US, the Weimar republic's inflation sprout before the Great War, the stagflation of the late 70's-early 80's, Zimbabwe in 2008 etc.
This time around, the 'problem' is of the reverse kind. This time around, it is deflation (or perhaps, more appropriately, disinflation) that is the issue. The 'poster child' of this malaise is Japan and its two decades of deflation.
One characteristic of the present configuration that is up to par with the historic experience (to a greater rather than to a lesser degree) is the presence of debt build-up. Just like in the Great Depression and before WWII in Germany, debt in today's economies is by all accounts too high (debt is 'too high' not because of its absolute volume-which is admittedly high too-but due to the systematically subdued ability to service it).
Another characteristic of the current state of affairs is the subdued returns to labor income ('subdued' when compared to productivity). This is mostly related to the US economy and for a period of roughly fifty years since the early 70's but other industrialized countries show similar patterns for about the same period. This is the outcome of a number of effects such as financialization and innovations related to communications (internet) and transportation (containerization) that increased capital mobility globally. To some, the present situation is the result of the success of what has been called the 'neoliberal agenda' (related to the 'Washington consensus').
A third characteristic of the modern global economy is what has been called by B. Bernanke the savings glut hypothesis. The former Governor of the Federal Reserve, made a case of why interest rates are subdued in the US (and the world). As the argument goes, there is an oversupply of savings that are looking for productive investments but cannot be accommodated by the current state of affairs. Hence, real interest rates (a measure of opportunity cost for real activities) will be low for as long as the conditions responsible for the oversupply, persist.
Perhaps the most interesting approach of the current configuration is one put forward by Brown's Watson Fellow M. Blyth. Blyth talks about economic regimes of changing institutions (a-la Aglietta). Juxtaposing 'the 70's' with the present, he finds a shift in institutional set-up, from an early one aligned with debtors' interests, to the later one aligned with creditors' interests. Regimes change when the underlying institutions become fetters rather than facilitators. Our present day experience is-according to Blyth-a (potential) turning point.
The preceding exposition was an attempt to frame the current configuration in a short and succinct way. As it is claimed in the introduction, inflation is dependent on the particulars of each historic (economic) epoch. During the present 'regime', low inflation is a strain rather than a strong point in the system. It makes debt repayment difficult. In turn debt overhang subdues growth by dampening profitability and allows for the continued trend of productivity-compensation gap. Low demand acts as a constraint on investment profitability prospects which in turn constraints growth prospects even more. This vicious cycle is but one example of why zero inflation is not a desirable level of inflation, at the current juncture.