I pretty much agree with the answer that Brian provided except for:
As for the effect of monetary aggregates, there are any number of reasons to expect that their changes will have no observable linkage to activity.
This is not a conventional view hold by mainstream economists, although to Brian's credit he points that out by saying:
Nevertheless, some economists still attach significance to monetary aggregates, one of them will have to offer an explanation.
Even though using 'some' as a synonym for mainstream of the profession is a bit strange.
Mainstream View on Significance of Money Supply:
The conventional view is that changes in money supply actually matter for what the inflation is. For example, this holds true for virtually all general equilibrium macro models presented in Advanced Macroeconomics from Romer which is classic graduate macro textbook. You will find the same views expressed in virtually any undergraduate macro textbook as well. However, there are two caveats:
- As shown by Krugman (1998) in his influential paper its not just money supply what matters. It is also the expectations of money supply that matters! Any expansion of money supply that is precieved as being only temporary and promptly reversed will have only limited impact on inflation and inflation expectations at the best.
- There are certain situations, a prominent example of such situation would be when interest rates are at zero lower bound (ZLB), where inflation becomes unresponsive to money supply. The reason for this is that at zero lower bound peoples willingness to hold cash becomes infinite as money themselves carry implicit zero nominal interest and hence if central bank would try to push for negative interest rates people would just soak up all the extra cash by hoarding it.
However, even despite the above caveats mainstream economists generally do believe that money supply affects inflation because ZLB is not a 'normal'state for an economy to be in - at least not historically. Moreover, also even though peoples expectations of money supply expansion can be fickle and central banks that are know as 'inflation fighters' might initially not have enough credibility in their commitment to permanently increase money supply, this obstacle is very situational and it is agreed by conventional economists that if central bank is really committed to monetary expansion the expectations eventually adjust.
For example, in 2019 the IGM forum- which is arguably the best poll for discovering what top mainstream economists think as it interviews a cross-section of top mainstream economists of diverse political beliefs, gender and age showed that in response to Q: "Countries that borrow in their own currency can finance as much real government spending as they want by creating money?" which is arguably a proxy to whether M matters for inflation, as the biggest problem with increases in M is that it leads to inflation, 26% disagreed, 57% strongly disagreed and 7% had no opinion and once the responses were weighted by confidence 24% disagreed and 76% strongly disagreed.
Furthermore, of course just because mainstream economists claim there is a relationship that does not mean there has to be one - that would be fallacy of arguing from authority, but my point here is to represent the conventional/mainstream view. Of course, mainstream academia is also not a monolith I am sure that there are mainstream economists who might hold different views but I believe it is fair to say that if they exist they are in minority.
View of the Bank of Canada (BOC) on Money Aggregates:
Furthermore, Brian misrepresents the official view of Canadian central bank presented on their website. However, I do not want to insinuate he does that on purpose because I think he is valuable user and even though I disagree with him here I actually hold him in high regard and I am sure he would never do that purposefully, rather I believe its due to erroneous assumption that absence of the series is evidence for their views on role of money supply which is fallacy.
In fact the same page that Brian provided link to BOC states:
For further discussion of uncertainty as well as the information and analysis used to inform monetary policy decisions at the Bank of Canada, see Jenkins and Longworth1 (2002) and Macklem2 (2002).
Hence the website does not offer all details. Now if you open both of those papers you will see that actually money supply matters:
For example in Macklem, Tiff, "Information and Analysis for Monetary Policy: Coming to a Decision." Bank of Canada Review, Summer 2002: 11-18 we will find:
The economic model used in the staff projection
focuses on the links from interest rates to spending by
households and firms. Information on various holdings
of money and credit provide yet another view of what consumers and firms are doing and planning to do. In
order to spend, consumers and firms need money or
credit, so the evolution of the monetary and credit
aggregates provides clues to spending plans. In practice, these aggregates are also affected by portfolio
shifts and other purely financial developments, so, as
with other high-frequency indicators, the challenge
for the staff is to separate the genuine signals about
economic activity and inflation from volatility related
to other factors. Regular contact with financial institutions provides useful insight into the particular developments that appear to be affecting the growth of
money and credit at the time. Information is also
obtained on credit spreads in bond markets and on
any changes in the conditions under which banks are
lending to businesses and households as indicators of
changes in credit quality and availability.
The staff in the Bank’s Department of Monetary and
Financial Analysis assemble this information to provide an overall view from the financial side of the
economy on the outlook for output growth and inflation, as well as on the risks surrounding this outlook.
Based on this analysis, they also make a recommendation to the Governing Council on the setting of the target overnight interest rate at the next fixed announcement date.
Next in Jenkins, Paul and David Longworth, "Monetary Policy and Uncertainty."Bank of Canada Review, Summer 2002: 3-10.
Second, it [The BOC] examines data on monetary and credit
aggregates, as well as information on credit spreads
and overall credit conditions. The purpose is to assess
the behaviour of financial intermediaries, the financial
conditions of households and of the business sector,
and the implications for demand and inflation pressures in the economy.
The emphasis and comments in  are mine.
Hence unambiguously BOC takes monetary aggregates into account and believes they have some impact on inflation. Or to be more precise it at least shows that they claim to do that (nobody can of course see what is in their heads).
We can only conjecture why BOC does not list money supply separately on the website but mentioned it in the papers. My best guess is that BOC does not list it because money supply is generally less useful for short-term forecasting, certainly not as useful as augmented New Keynesian Philips curve for example (which is based on output gap that is listed there). However, forecasting ability of a aggregate has no implications for underlying relationships. Even nonsense variables can be useful for forecasting because they happen to co-vary with the variable you want to forecast in some way and vice versa - this is well accepted in literature on forecasting.
For example, according to Economic Forecasting and Policy from Carnot, Koen and Tissot certain types of forecasting models are built:
... models are based solely on the statistical properties of the series under consideration, irrespective of any interpretation or causal relationships informed by economic theory...
The authors also state that such models can often provide relatively better forecasts than the structural (i.e. theory/underlying economic relationship based) models. Also you will find this caveat in pretty much any text on forecasting even outside economic profession. Hence whether money supply is used for forecasting or not in itself tells us nothing about what the actual relationship is or whether institution using such forecasts thinks what the actual relationship is (also note forecasting should not be confused with empiricism or statistical estimation of some empirical models).
However, it is actually a common mistake that even many very intelligent people make all the time to assume that if there is some fundamental relationship between two variables they should be good at forecasting or vice versa. Nonetheless, any such assertions are simply just a fallacy and as a consequence one cannot generally judge underlying economic relationship simply from forecasting models central banks/professional forecasters use let alone from the variables they use for forecasting.