Shallow depth of field (DOF) and background blurring happen from a number of factors. They are (in order of highest importance to least):
- Subject distance (the closer to the subject you are, the more the
background will blur)
- Subject-to-background separation (the farther from the background the subject is, the more the background will blur)
- Aperture setting (the wider the aperture, the thinner the DOF)
- Focal length (the longer the lens, the thinner the DOF)
Simply setting a typical 1/2.3"-format sensored P&S camera on macro mode doesn't give you background blur. It just changes where the lens searches for focus to a nearer range. It's having the camera in very close to the subject that gives you the blur. Shooting in macro mode at a subject that's not within the macro distance simply means you won't be able to focus correctly.
Larger sensors typically yield thinner DoF because of factors 1 and 4. The larger your sensor, the longer the lens will be, and/or the closer to your subject you'll get a similar framing than with a smaller sensor. And both of these will increase the background blur.
Typical small P&S cameras are handicapped from giving thin DoF because they have small sensors, tiny lenses, and small maximum apertures: all of which increases DoF. This is a feature, not a bug. Most P&S shooters don't want to have to worry about getting things out of focus. Bokeh is out-of-focus blur.
So, the specs you really want to shortlist for shallow depth of field are:
- Sensor size -- Bigger is going to better for this; and you'd probably want at least 1"-format, if not APS-C.
- Lens maximum aperture -- Bigger is going to be better. The maximum aperture (or max. aperture range across the zoom range of the lens) is the f-number(s) given after the focal length on the lens. The smaller the f-number, the bigger the aperture settings you can use. f/2.8 is probably your minimum--you'd really love something that's f/1.4.
Probably the best "P&S" (by which I mean a fixed-lens compact camera) today for thin DoF is probably the Sony RX-1. It sports a full-frame sensor (1x crop), like a Canon 6D or a Nikon D600, and has a 35mm f/2 Zeiss lens on it. It costs a bomb.
APS-C (1.5x crop) fixed-lens compacts would include the Fuji X100 series, Nikon's Powershot A, and the Ricoh GR. The Canon Powershot G1 X series is slightly larger than 4/3"-format (2x crop). Sony's RX-100 is a 1"-format (2.7x crop) compact. And Fuji makes two 2/3"-format (4x crop) compacts (X10/X20, and the XS-1 bridge camera).
All of these cost a bomb but are less than the RX-1. $500-$1000 is the more typical price spread on these cameras. Be aware, however, that the lenses on these cameras may or may not zoom. And, of course, this is an area where camera technology and manufacturers are constantly churning out new models with different specs. Expect all these models to be different in a few years; just as this answer is updated from the ones that were made preivously. A lot has changed in the camera landscape over the last three years.
You could also go for a mirrorless camera (Sony E-mount, Fuji X, micro four-thirds, Samsung NX, etc.) or a dLSR and have the freedom to change lenses, which will typically allow you access to wider maximum apertures, but system cameras tend to cost more because you have to buy the rest of the system as well.