Most don't like images with motion blur. An image out of focus is often seen as just a poor photograph. However, I really like motion blur. I've experimented with motion blur, and whenever I'm out on a photoshoot I try to take a few shots with slower shutter speeds. Motion blur conveys motion (I know I'm stating the obvious) and the activity of life. However, done poorly and you're right, it's just an image out of focus.
The key to motion blur from my experience is to focus on a subject (or area in the scene) and then lock down AF. Shutter speed needs to be slower than 1/50 of a second in my experience. At night that is rarely a problem. During the day you may need to switch your camera to Speed priority or Manual. I always try to keep ISO as low as I can, but that's not always possible if you shoot at night like I tend to do. Whenever possible I set my camera to 100 ISO, and take a few sample shots. Sometimes I use auto-ISO at night and almost always use a tripod.
This image was taken on a really busy night. I had set up my camera on a tripod to take some street scenes. These 3 guys walked up and were just watching the activity pass by. I decided to focus on them with my camera set at F/4.5 so I could get the background blurry (in addition to the motion blur). ISO set at 400 with my Nikkor 18-135mm lens set at 38mm. Speed was 1/40 of a second.
This image of a young lady's backside was taken during the day. I had my camera on a tripod about 25 feet back from an intersection. I used a cable shutter release to minimize camera shake. This may seem a waste since the idea was to get a blurry picture. However, my goal was to get motion blur but as sharp of focus as possible on my subject. I set my camera at F/20 for a deep DOF, ISO 100 and my 18-135mm lens at 85mm. Speed was manually set at 1/20 of a second.
I took this photo last night while out on another photoshoot. I was packing my kit in my car across from a highrise. A couple were walking by across the street (they had just been arguing about something?) and I took a shot. They were a bit tipsy so they must have been hitting the clubs. I wanted to capture an image that conveyed the sense of unbalance. I had a Nikkor 70-300mm F4.5-5.6 VR mounted. This photo was shot at F/4.5 ISO at 200 and the focal length at 102mm. Speed was at 1/10 of a second which produced significant motion blur.
OK, these images would NEVER win an award, or end up on any one's must have list, but that's not the point. Shoot what you like, and I like motion blur.
From my experience if you want to get motion blur during the day you will need to set your camera to Speed priority. At night you can get away with Aperture priority. Speeds below 1/50 of a second will start producing motion blur. Below 1/10 of a second and I've found there is too much blur. Try to set your ISO as low as possible, but if you are a night owl like me you may need to use a higher ISO setting or auto-ISO. Anyway, shoot away ... I know I do.