We spent a few weeks in Melbourne last year having a squiz, with a local yocal.
We felt Northcote, Thornbury, Fitzroy North, Clifton Hill might be ok. Many parts are still tacky though.
Coming from Brissie, we feel more comfortable in Melbourne's leafier inner east. It is a no brainer that the demographic is more recession resistant there.
You might get a house in Thornbury. We also looked further out around Preston, and some parts didn't seem too bad.