I understand that this answer will vary depending on neutral and belligerent and year, and kind of expect multiple answers depending on that. I am trying to get an idea of how much neutral shipping there was during WWII, and what neutral merchant marines were doing during WWII. Stories of any particularly risk-taking merchants are also appreciated!
My assumptions and specific questions for each significant merchant marine neutral are as follows:
Sweden, Finland: I assume they would be stuck trading with Germany in the Baltic. Would a merchant attempt to run a shipping route to the UK or US, or was that trade halted by Germany?
Central/South American countries: I assume they would mostly trade with the closest and safest belligerent, the US, plus maybe Australia/New Zealand and UK possessions in Africa and India. Was there much risk from German U-Boats for this trade on neutral flagged ships?
Would a Central/South American merchant attempt to trade with Japan or Germany/Italy? Vichy France?
Ireland: Are they stuck mostly in the US/UK North Atlantic convoy trade? If alone in the North Atlantic, would an Irish flagged merchant be attacked by German U-boats? Would an Irish merchant even attempt to trade with Germany? How about Vichy France? Japan?
Spain, Portugal: This is an interesting one. I know these countries were friendly with the Axis powers, and could provide neutral ports through which Germany could acquire war resources (via rail transhipment). So where was their merchant marine trading? How much of their ocean-going import trade was eventually bound for Germany/Italy? Also did Spanish/Portuguese merchants attempt to trade with Japan?
Turkey: Another interesting one. How much of their merchant marine was trading with the UK/USSR versus Germany/Italy? How much did a Turkish merchant ship need to worry about British or German submarines in the Mediterranean? Would Turkish merchants be allowed through the Suez Canal during the war? Would Turkish merchants try to trade with Japan? How about the US? Was Istanbul used as a neutral port for rail transhipment to Italy/Germany?
Vichy France: Not technically a neutral, but interested in what their merchant marine was doing 1940-1945. Were they even allowed to trade with the US/UK? Could they trade with neutrals without interference from the US/UK? Also interested in the status of the French merchant marine in the Far East, with regards to trading with Japan.
All neutrals: Were any incentives provided by the US/UK for neutral merchants to join the North Atlantic Convoys or USSR lend-lease shipping? If so, did any neutrals take up that offer? Did Japan/Germany/Italy provide any extra incentives to try to entice neutral merchants to trade with them and/or not trade with the Allies? What efforts did the Allies put into stopping neutral trade with Axis powers?
For Sweden, about half the Swedish merchant fleet of some 1,8 million tons were caught outside the North Sea blockade enforced by the British and the Germans after the German invasion of Norway in April 1940. The other half was in the Baltic Sea, the Danish Belts and the area east of Dogger's Banks. 13 Swedish merchant ships were sunk from April to June in the fighting, and by the end of it, Sweden had been cut off from foreign trade except at the mercy of both Britain and especially Germany.
Sweden did do limited trade with other neutral nations throught "lejdtrafiken", an agreement with both the British and Germans to pass a small number of ships through the dual blockade - the first ship, the tanker M/S Sveadrott carrying 13 700 tons of oil from Venezuela arrived on the 8th of September, 1940.
"Lejdtrafiken" proved vital to supplying Sweden, and 5 ships per month were allowed through 1940 and 1941, which was increased to 70 ships per eear in 1942 and 80 ships per year in 1943. As these ships brought in vital strategic resources, chiefly oil (usually from Mexico or Venzuela), rubber (mostly from Brazil), grain (mostly from Brazil), meat (mostly from Argentina) and colonial goods such as tobacco, cocoa, dried or canned fruit, coffee and tea. The latter were usually the cause of small celebrations when the arrival of a ship was announced in the papers, as the schedule and route of the ships were kept a strict secret according with the agreements with the British and Germans.
As you can probably guess, with the fixed number of ships allowed to pass, Sweden used the largest, fastest and most modern ships for this trade to get as much goods as ever possible - despite this, "lejdtrafiken" never managed to bring in more than about 15% of Sweden's pre-war imports.
In October 1944, the trade was closed due to Sweden stop on export on iron ore to Germany and Norwegian ships in Sweden breaking out into the North Sea to sail to Britain with a load of precious ball bearins. It was resumed in early Spring 1945.
10 ships were sunk and 166 Swedish sailors died during this trade during the war - 5 by mines, 4 by German subarines (usually while sailing past Britain) and 1 by German planes.
The ships caught outside the German and British blockade usually sailed for the Allies - partially becuase those were shippping that paid well, partially because there were little other things to ship. The dollars and pounds recieved in payment was used to pay for the imports that were brought in through "lejdtrafiken". The American State Department also chartered the passanger liners M/S Gripsholm and S/S Drottningholm - I /u/vonadler have written about this a bit here - to exchange civilians and diplomatic staff between the warring countries. Of these ships, 270 were sunk during the war, and roughly 2 000 Swedish sailors died.
The ships in the Baltic Sea maintained regular peace-time trade in the Baltic Sea - they mainly shipped iron ore (alhough wood, paper pulp, copper and ball bearings were also shipped) from Luleå (in Summer) and Oxelösund (in winter) to Rotterdam, Hamburg, Emden, Bremen and Danzig/Gdansk and returned with coal, coke, artificial fertilizer, oil cakes (animal fodder) and artificial fuel and rubber. 12 ships were lost -2 fishing vessels sunk by German minesweepers, 1 of a Soviet mine outside Memel, 1 by a British mine, 8 by Soviet submarines. 187 sailors and passengers died, and 6 sailors picked up as prisoners by a Soviet submarine are executed at Kronstadt.
So, what you do depends first on wether or not you were caught outside the German and Britihs blockade of the North Sea. If outside, you would most likely sail the dangerous trade for the British and later the Allies through the Atlantic, delivering vital war supplies to Britain and braving German submarine attacks, mines and air attacks, gaining vital foreign currency for Swedish imports through "lejdtrafiken". If your ship was modern, large and fast, it could be selected to be one of the ships going through "lejdtrafiken" bringing in vital supplies for the survival of the Swedish people and the Swedish armed forces (and in case of coloinial goods, morale).
If your ship was in the Baltic Sea or along the Swedish west coast when the blockad was erected, you would sail iron ore and coal trade between Sweden, German-controlled Poland and Germany or bring in some other goods. This trade as a bit less dangerous, but Soviet submarines in 1942 and 1944 still took its toll, especially the sinking of the passenger liner S/S Hansa, which was sailing its regular route between Nynäshamn just south of Stockholm and Visby on Gotland on the 24th of November 1944, when 84 people died.
The chief incentive to sail for the Allies across the continent were the payments in pounds and dollars, which could be used to purchase vital food and raw materials in neutral countries to ship back to Sweden through "lejdtrafiken".