The factual basis for speculation that the dollar is losing seigniorage to the euro turns out to be weaker than we thought, according to a European Central Bank study reported by the Financial Times:

Since the introduction of euro notes and coins in 2002, circulation has soared above expectations. Last year, the value exceeded the value of dollar bills in circulation.

However, the ECB's study suggests that global demand for the euro is less important than previously thought and that, beyond Europe, the dollar remains the cash of choice.

Net shipments of euro notes to outside the eurozone are declining, says the report. Only 10-15 per cent of euros in circulation are held abroad. About 60 per cent of US dollars in circulation are held overseas.

The study found that Europeans carry around more cash per capita than Americans, and (as a standard inventory model would consequently predict) withdraw more per ATM transaction. The FT offers as an explanation that credit and debit cards are less commonly used in Europe. I’d say that’s jointly determined with cash-holding practices, not exogenous, and therefore part of what needs to be explained rather than an explanation. (I don't think Europe is lagging the US in technical knowledge of how do do card payments.) For an exogenous explanatory factor I would suggest: marginal tax rates are higher in the EU, so the underground cash economy is relatively bigger.