We only have one account, it kind of covers everything. We RARELY have any sort of event that requires us to dip into the fund. We have wiggle room in the budget, and if something comes up we cashflow the expenses and send less to savings that month.
I do keep mental tabs on likely expenses and attempt to budget for them. In your example, you mentioned a major home repair (roof or furnace) and a needing a new car. I don't think any of these things is likely to be a true emergency, with the possible exception of the furnace. If you do regular maintanance on your car, you should know if it is in good running condition or if it's likely that it will need to be replaced soon. A new roof shouldn't be an emergency either- if you keep an eye on it, you'll see when it is starting to degrade and you should be able to plan accordingly. If a storm comes through and rips off the roof (a true emergency), that should be covered by homeowner's insurance. As for the furnace, a major malfunction in the middle of the winter can be problematic- it happened to us last winter. While needing to buy an entirely new furnace could be several thousand dollars, the most we've ever paid to have a furnace repaired is a couple hundred dollars.
I guess what I'm trying to say is that with most major expenses, you CAN plan ahead and budget accordingly. A $10K sink fund would be more than sufficient for US, but if you don't think it's enough for you, then add to it