I don't really have a designated emergency fund, but in the event of an emergency, I would tap funds in roughly this order:
1. Credit cards
2. Temporary funds in my checking account
3. Taxable investments
4. 401K loan
5. Apply for a short-term bank loan
6. Roth IRA principal
7. Roth IRA gains
As for what is "best", you need to judge that for yourself. I happen to think that keeping a lot of money in cash doesn't make much sense from a purely financial standpoint, but I'm not going to judge people negatively if keeping months worth of cash gives them a warm fuzzy feeling inside. I've said it many times in this thread, but the point of investing is improve your life satisfaction, not to run up the score on your balance sheet.
OK, I looked up Vanguard Total Bond Market, and the drop from peak to trough in 2008 was 7%. That is assuming you bought at the worst possible time and sold at the worst possible time, and it ignores the dividend yield you would have earned in the interim. And furthermore, you could probably end up getting that 7% back eventually via tax loss harvesting.
It doesn't seem to me that an emergency fund that is 7% larger is hugely more robust to emergencies. And more to the point, it only takes 2 years for the bond-based fund to have already earned 7% more than a cash fund would. So the scenario where the cash fund performs better requires incredible bad luck - not only do you need to have a major emergency in the next two years, but it also has to coincide with you happening to both buy and sell at the worst possible time. And even then, it is not winning by much.