Grease is usually 'out' unless specifically called for. It's handy for situations favouring a sticky semi-solid lubricant, such as power gearing. It's also good for preventing corrosion because it doesn't drip.
Generally, lubrication has tended to move away from high-maintenance systems needing men with oil-cans and grease guns to pumped and sealed-for-life systems. In broad terms oil has tended to replace grease as the preferred lubricant.
For lathes and machines with slide-ways, especially vertical slide-ways it's a good idea to use 'Way Oil'. Being sticky, it's not pushed out of the way by the saddle. Personally I don't bother, instead using ISO32 Hydraulic Oil for almost everything. Avoid certain EP oils if your machine has bronze bearings or any brass parts : the chemical added to improve pressure performance attacks Copper. The label carries a warning.
3-in1 is a general purpose household oil good for door hinges, garden tools and general cleaning. Not particularly useful in a workshop I feel, and – like WD40 – deadly to clocks. 3-in-1 (and similar concoctions) contain vegetable oil that gums up over time and collects grit. This murders fine mechanisms like clocks by stressing and wearing the mechanism. WD-40 commits the opposite sin; it contains a light oil that washes out the pivots. At first the clock is lubricated by a thin layer of oil left behind but this soon evaporates leaving a sticky water-repellent layer that has much the same effect as 3-in-1, except the bearing is run dry.
Always use the correct type of oil on a clock or similar mechanism; workshops tools are far less fussy.
Is there a particular reason you favour synthetic oil? I suggest the particular specification and purpose of the oil is more important. Most Synthetic oils are formulated to improve their performance at high temperatures. This is rarely needed in the workshop and the additives might be positively harmful. Good old-fashioned oils – like plain ISO32 – are more appropriate. Although I tend to avoid Motor Oils because they contain additives suited to boiling hot engines rather than cool lathes, of course any oil is better than none.
Dave