How does smartphone optical image stabilization compare with DSLR lens stabilization?

You have to distinguish between software and hardware image stabilisation, often referred to OIS - Optical Image Stabilisation - for the hardware kind, where the lens or sensor moves to counteract camera shake and perhaps just IS for the software kind, where software techniques are used to remove the effect of shake. However, Canon's stabilised lenses have traditionally been simply labelled 'IS', despite being a hardware-based optically stabilised system implemented in their IS lenses.

There are 2 ways to implement IS in hardware - move the lens elements or move the sensor. With in-lens IS, every lens has to have the IS capability; with sensor based IS, the IS hardware is in the camera and benefits every lens fitted. As far as I am aware, all phone cameras that implement IS have sensor-based systems.

Software-based IS only works (I think) for video and attempts to identify a consistent part of the image and keep it aligned from frame-to-frame, cropping the image slightly. This can be done as the video is recorded or afterwords in video editing software. The former method required that the camera implements software IS; the latter doesn't.

Traditionally, stabilisation for D-SLRs has been lens based and was mainly seen on Canon (IS - Image Stabilisation) and Nikon (VR - Vibration Reduction) lenses. Now most lens manufactures offer stabilised lenses whilst body-based stabilisation is becoming more common, although not by Canon and Nikon, who stick with a lens-based solution.

Proper hardware based IS is just becoming more common in phones, and is exclusively sensor based, but the systems are largely equivalent.