Learn English – “I currently work at” vs “I’m working at”

There can be a subtle difference.

If I were creating something, e.g. writing some new programming code; putting some electrical or mechanical components together to achieve a particular objective; installing and setting up a new TV set for the first time ^[1]; then, if unsuccessful, I might say "I didn't get it to work" or "I can't/couldn't get it to work" ^[1]. (It had never worked, because it was new.)

If I were repairing something that had previously worked (e.g. repairing a previously working TV set ^[1]) and the repair was unsuccessful, then I would probably say "I didn't/can't/couldn't get it working". (It had previously worked.)

Note that these are subtle differences, and I do not mean to imply that the expressions and scenarios are necessarily always used that way around, not that it would be wrong to use them the other way around.

_{^[1] Thanks to Janus Bahs Jacquet, whose comments contributed some examples and other input to my revised answer.}

"Start work" is the idiomatic expression (at least in AmE) for showing up at your place of employment. You may or may not do anything once you arrive. "Start working" means initiating some endeavor. Thus: "I start work at 9A, but I don't really start working until 9:30." – deadrat

My feeling is that for a job, the more natural choice is "start work." A new hire would respond to a question about a start date at the company with "I'd like to start work as soon as possible." A building contractor would make the same choice when talking to a homeowner about a project. A general activity would take "starting"+on: "Cleaning out the basement will be a chore. I need to start working on it soon." I doubt I could document that as a general rule, and remember I speak the American kind of English. – deadrat