Polls show that the public thinks it likes low inflation. But it doesn't. Thinking you like low inflation is reasoning from a price change, which is almost always a bad idea. Thus we should be very skeptical when the public claims to like low inflation. In fact, they like low inflation only when delivered via rising aggregate supply. The public absolutely hates low inflation that is produced by falling aggregate demand (i.e. tight monetary policy).
Here's a graph showing the results of various public opinion polls on the economy:
As you can see, the public's view of the economy fell sharply during the first half of 2008, as an adverse supply shock drove inflation higher. But by early 2009, inflation had fallen to roughly zero, the lowest level of my entire life (since 1955.) And yet the public's view of the economy continued to fall to extremely low levels.
Some would argue that I am not holding other things equal---unemployment was rising sharply during early 2009. That's true, but unemployment was rising sharply precisely because tight money was driving inflation down to zero. (It would be like saying the public doesn't mind falling out of 100 story buildings, just hitting the ground.) The public may say it likes low inflation, but it behaves as if it likes low unemployment. That graph tracks real GDP growth and the unemployment rate far more accurately than the inflation rate. Growth fell and unemployment rose in 2008 and early 2009, as the public's view of the economy was getting worse. Then it gradually improved as unemployment fell back and the economy recovered. In contrast, inflation today is higher than in 2009. As it rises, the public's view of the economy has improved.
In my view, there's no such thing as "public opinion" when it comes to inflation. To the public, the term 'inflation' means something very different from what it means to economists. Surveys showed that in 1990 most Americans thought inflation was higher than it had been 10 years earlier (it had actually fallen from 13% to 4%). Perhaps they confused "inflation" with "cost of living". Recently, inflation rose to 2%, and yet the public's view of the economy is now more positive than in many years.
If a central banker were to read a public opinion poll showing that people opposed inflation, and then decide to reduce the inflation rate to zero, they would be in for an unpleasant surprise. Whatever you think about "public opinion" and inflation, keep in mind that the public absolutely hates an economy with zero inflation produced by tight money. We saw that in 2009, and we'd see it again if the Fed suddenly drove inflation down to zero.
I think the public is also sensitive to the fact that bouts of inflation seemingly hurt the most vulnerable. Those unemployed and/or on some sort of fixed income that is not inflation-adjusted.
Actually it's the unemployed who are helped the most by inflation, if it is produced by monetary stimulus. (Of course supply-side inflation hurts almost everyone.) Demand side inflation tends to generate jobs for the unemployed. As far as the "vulnerable" people with fixed incomes, they are typical not poor (AFAIK). The poor tend to rely on programs like Social Security, which is indexed to inflation. People with unindexed annuities are likely to be at least middle class.