°µÍø½ûÇø

Visa and taxation rules depend on where the work is performed - in Singapore or abroad. It does not matter in this respect where and how you are paid.
If your Singapore employer sends you to work outside of Singapore temporarily (e.g. on a business trip), this is seen as incidental to your Singapore employment and there is no change in visa and taxation.
If you are sent to work abroad for longer, e.g. on secondment, the salary for that work is not taxable in Singapore and you do not need a Singapore work visa (your employer should check with MoM whether you can keep your existing work pass or it should be cancelled and a new one applied for when you return).

I had related constellations twice in my career:
- Once I was seconded by my Singapore employer to a project in Indonesia for 6 months. I kept my EP, but did not have to pay Singapore tax during that time. (I hat an Indonesian work visa and was taxed in Indonesia, which charges more - luckily my company covered the difference, so I suffered no loss.)
- Once I was employed by a Singapore employer to exclusively work in Taiwan. Although I was paid in Singapore, I did not need a Singapore work pass and was not taxed there either. (I got a Taiwanese work visa - but, luckily for me, Taiwan taxed only income paid in the country, so I paid no tax anywhere. This loophole has probably been closed by now.)

You mentioned about temporary and longer duration business trips, what about when neither is the case, I mean if the employee wants to work remotely from another country for personal reasons while on EP, is that legal to do so?

Of course you can, at any time, terminate your employment in Singapore (by resigning or mutual agreement with the employer) and start working in another country (for the same or another employer).
You are then employed (or if the employer is not located there, freelancing) in that other country. Ask about visa and taxation rules in that other country forum!
If you are sent by your current employer, you can in some cases (as I wrote before), keep your EP. But it is not necessary to have one, from a legal standpoint.