The municipal-bond market has been handed a bullish gift. Now they just have to figure out what to do with it.

The Federal Reserve signaled last week it wouldn’t raise interest rates in 2019, triggering a rally in state and local debt because interest rates and bond prices move in opposite directions. At the same time, record-breaking amounts of cash are coming in: municipal-bond mutual funds have attracted more money in 2019 than they have during the same period of any other year since 1992, when records began, according to Lipper US Fund Flows.

All of that has led to...