In Progress 59 mission, the service module was damaged and venting pressured gas, which caused the spacecraft to tumble.
What could be done if similar situation occurs on a Soyuz?
Or generally, What abort modes does Soyuz have in space?
I'll start with a general overview of the abort modes available to Soyuz (including before getting to space for completeness).
In the early portion of the mission and while sitting on the Launch Pad, the crew can be evacuated by the use of the SAS Launch Escape System. On top of the Soyuz Rocket is a Launch Escape Tower that is used for Launch Aborts prior to liftoff and during the first 157 seconds of the flight.
At T+157 seconds, the SAS and Payload Fairing is jettisoned. After that, a variety of abort modes are available for the Soyuz Spacecraft. Depending on the abort scenario, different mechanisms are used to separate the Soyuz Capsule from the Rocket. After separation from the Launcher, the Soyuz conducts a commanded Module Separation and the Entry Module uses its attitude control system to place the vehicle in the correct entry attitude.
The abort trajectory that takes the Soyuz to a landing point somewhere downrange the ground track, largely depends on the timing of the failure. The later an abort occurs in the ascent, the higher the G load the crew and vehicle experiences. For an abort 400 seconds in the flight, up to 21Gs occur on the way back to Earth. After Entry, the Soyuz performs nominal landing operations to bring the crew back to Earth. All abort scenarios are considered to be survivable by the crew.
Emphasis added by me. Also, for any Kerbal Space Program fans who were wondering, SAS is the transliteration of the Cyrillic САС abbreviation of Система Аварийного Спасения, which means Emergency rescue system.
I'm having a rather difficult time finding details on the "different mechanisms used to separate the Soyuz Capsule from the Rocket", but Soyuz 18a actually did this:
At T+295 seconds, the deviation was detected by the Soyuz's guidance system, which activated an automatic abort program. As the escape tower was long gone by this point, the abort had to be performed with the Soyuz's own engines. This separated the spacecraft from the third-stage booster and then separated the orbital and service modules of the Soyuz from the re-entry capsule.
This book (on page 70) references another document and indicates that before T+522 (which is shortly before third stage burnout) the orbital and descent modules should separate from the rocket and go into a controlled descent, while after T+522 the entire Soyuz vehicle should separate and go into a ballistic reentry. Unfortunately it doesn't describe how the separation from the service module would be accomplished and the book goes on to comment that Soyuz 18a aborted before T+522 but still had the entire vehicle separate.
I suspect I'd have better luck finding more information if I were searching in Russian, but my Russian isn't that good. In the absence of additional sources I'm going to speculate that splitting the Soyuz vehicle in an abort before T+522 depends on the third stage booster being shut off, leaving the descent module in a suborbital trajectory, while Soyuz 18a was forced to use the Soyuz engine to escape from the third stage booster. Furthermore, after T+522 the trajectory is either orbital or doesn't reenter in a convenient location so the Soyuz engine has to be used to ensure a proper reentry.
What we know about Progress 59:
The spin wasn't as dramatic as the one revolution per second spin of Gemini 8, which was manually corrected for by the astronauts on board. In the absence of a system failure causing the RCS system to not work I'd say the occupants of a Soyuz vehicle could have countered the spin without a problem.
Loss of contact with the ground isn't as critical with a manned vehicle as with an unmanned one - the astronauts/cosmonauts on board could initiate the abort sequence where the automated ship has to be commanded remotely when something goes wrong.
Assuming that wasn't damaged, Soyuz has life support for 30 man-days (so 3 people, 10 days). That would have been enough time to last until the vehicle decayed.
The breakup (probably from an explosion) and multiple system failure would have been the biggest issue.
If the engine was damaged or unable to fire (as is suggested by loss of pressure in the fuel lines) then the ability to use the vehicle to initiate deorbit is in question. Without the main engine perhaps RCS thrusters could be used to help decay faster, but I don't know how much delta-v they have available or how much they would need.
If the explosion suggested by the breakup event damaged the service module beyond just the engine that could impact life support or prevent the onboard controls from working. All we know is that multiple systems failed, the specifics of which ones would impact what could be done to abort successfully.