It's going to vary wildly depending on your situation. Single vs. couple vs. family, apartment vs. house, LCOL vs. HCOL, etc. I think it's hard to generalize.
Expenses to keep in mind:
The actual move - whether you rent a truck and do it yourself, or hire movers.
A trip to scout out the new area and find a home - flight, hotel, etc.
Three months rent for first/last/security deposit.
New things to fit the new space, e.g. rugs, curtains, etc.
Replacements for miscellaneous things people often toss out while moving, e.g. shower curtains, garbage can, toilet brush, etc.
Realtor fees (this is pretty rare if you're renting, but depending where you move it could be a factor).
Many of these things will vary heavily based on how far you move and how expensive an area you're moving to... 20x rent could be far too high (in my case it was more like 6-7x rent) or far too low (if, say, you moved very far to an area with very cheap rent). And of course your own behavior will change it too, like whether you DIY the move or hire movers, or if you buy a lot of new stuff or not, or sleep in a cardboard box for a weekend or stay at the Ritz... ;-)
