I have 6 mo of expenses in I-bonds, but I think the answer is "it depends".
To @Khatera's point, it matters what is meant by "emergency". I fully respect pulling out $$ when storms threaten, I've done the same when ice storms are predicted and that was also very useful. A few hundred for towing, et al, I would suspect people would have in checking/demand savings accounts... or at least I do. I don't want to mess with selling investments just because the budget was off $400 in some given month.
The "layoff type" scenarios of how to handle 6 months of expenses then can allow for a week or two to access the money, IMHO.
Then we have when someone made the decision, as well as needing to understand how interest rates work.
So let's back up and give a framework for @SwordGuy to get a handle on I-bonds.
If we look at the impact of inflation over time, then we find that "cash" investments (think CDs), tend to return very little after inflation (on a "real" basis). There are periods where they are negative real return, some periods with small positive return.
Let's tell a story for a moment with a 3% CD rate for 5 years. We can actually separate that 3% nominal rate into two pieces: Assumed inflation and the assumed real return. If we think inflation will be 2%, then this will be a real return of just under 1%. However, the CD purchaser bears the risk that the 2% assumption is wrong.
Now let's look at the I-bond. There are 2 forms of return to I-bonds, a fixed rate and the "variable rate" for inflation. The fixed rate is a "guaranteed" real return (two additional points on this guarantee). The variable rate is based on actual inflation.
Therefore, buy purchasing an I-bond rather than a CD, we have transferred the risk of the inflation assumption being wrong to the seller of the I-bond (because the I-bond purchaser gets return based on actual inflation). What we have then taken on is that our own personal rate of cost increases is different than the CPI-U (urban CPI) used as the basis for the I-bond. That's fairly nominal for an average urban individual, but does have some quirks for Mustachian folks and rural folks. To know what that is requires you know your own personal rate of cost increase :)
To me, the risk transfer is key to evaluating a CD vs an I-bond for an "emergency fund". A purchaser of the I-bond has transferred risk to someone else. To me this matters, because the point of an emergency fund is to minimize risk...
So it's time to ask what we mean by "emergency fund". We all know CDs and I-bonds are rarely good for the long term. That's why we see folks pointing at VSTAX for long term investment. But an "emergency fund" is working to deal with the "tail events" (when things go really bad for us _personally_), not a statistically analyzed long term return. Note that I'm not saying here we can predict CD/bond returns or that we even care to, but it's about transferring risks to others, so we have less factors of variability when things go wrong.
Given that, we should think about when "tail events" are likely to happen - and I would assert they tend to be correlated - meaning when the economy goes bad, we are more likely to see layoffs (see 2001, 2008). So for an emergency fund, the more stability we can have and the more risk we can transfer to someone else, the better.
If we assert the emergency fund is trying to minimize risk factors, then it follows that taking principal risk is off the table, so you're stuck with federally backed instruments: I-bonds, very short term treasury notes, or insured CDs and the like.
If you don't have that assertion, then yes, some folks choose to be all investment (VSTAX and the like) and just assume they can use HELOC, credit card or whatever for emergencies. That is a risk analysis for each individual (and not putting the key assumptions on the table is why folks end up talking past each other on this topic). To my point above on risk and in response to @trollwithamustache, I'd note that Muni bonds are an investment, not a place for emergency funds unless you are just as happy putting it in VSTAX. They can vary widely in value, so while they are somewhat liquid, you can't guarantee anything close to good returns when things "go bad". I picked up some crazy values during the '08 events when folks were dumping them. They are also not default-free, see LA, Detroit, Jefferson County Alabama, and others. So it's not that you *can't* do it, it's that you should understand the risks you are bearing.
Given all that, and thinking about risks, we can then evaluate cash vs savings accounts vs I-bonds. All are effectively equivalent guarantees. Cash is clearly resilient to power outages, et al, but comes with higher theft risk. I-bonds have some startup risks (the first year withdrawal restriction and 3 months of lost interest), but have transferred off the inflation risk. There is at least two other steps (sell, then transfer to bank account) beyond a savings account. Whether that is worth it depends on the immediacy of the emergency one is planning for and the risks an individual is willing to take.
Now back to I-bonds. IMHO, the fixed rate matters a lot to whether they are worth it. If there is no [or nominal] fixed rate (the situation for some years now), then it is likely that fully nominal interest items (CDs, etc) are better net return over time (because they will have an "assumed inflation" as a component of the nominal rate whereas the I-bonds are *guaranteed* to have zero). For my I-bonds, the fixed rate is 1.4%. I would not personally advise anyone to get them now when the fixed rate is 0%. (Hat Tip to @Gin1984). IMHO, there are better options... but if they go back to a higher fixed rate, then they are something to consider because it allows you to offload the actual inflation risk.