On popular days (Fridays, Sundays), expect Thays to have no pass holder seats left and the TGV maybe (!) a few few 20€ seats if you try to book on the day of travel.

On very popular days (Easter weekend, Ascension day weekend, start of holidays in France etc.), expect Thalys and TGV to be fully booked weeks in advance.

You can check for yourself here for Thalys and here for TGVs in France.

Thank you very much for you quick and clear answer!
Will the chances be higher for successfully making a reservation if I travel on a first class interrail ticket?

Maybe. I don't have statistics on that but my impression is that passholder seats are indeed fully booked earlier in 2nd class.

A Plan B route can be the reservation free Intercity to Amsterdam and then one of the few Belgium - South of France TGV´s usually one or two of them stops in Marseille :)

In these TGVs from Brussels, the pass holder seats sell out even faster than in Thalys :/

Better go to Lille with a change in Antwerp and take a domestic TGV from Lille. At least, these can be booked until there actually full (for 20€).

You can also travel in 1st class with an Interrail pass in 2nd class. Then you have to buy an upgrade ticket.

Example : Lille-Marseille, 40 euros.

Hi Joostvangrinsven, We encourage you to book Thalys and TGV reservations as much in advance as possible as these trains can get very crowded, especially around summer time. Cheers,

It seems that I only can check availability of seats after I have purchased an interrail ticket (in the phone app or on the website). And after I purchase an interrail ticket it might be very well possible that there are no seats left on the route and day I want to travel. A bit of a catch 22. Or am I looking at the wrong place? Is there a solution for this problem?

It seems that interrail is reluctant to show pass holder availability before you've bought a pass, which I also find disappointing.

For some trains though you can check availability via other websites, e.g. see my first reply.

@rvdborgt Thanks again for the quick reply. I checked the websites you suggested in your first message. At first I thought I had to enter an Interrail pass number, but it appeared that I only had to enter a pass type (1st or 2nd class). So now I am a step further. I see some trains appear on these sites (but not all of them?). 
I wanted to travel from Amsterdam to Marseille with a minimum of transfers. I can travel via Bruxelles Midi and that gets me to Marseille with only one transfer. But availability is low. Maybe I have far more chance to travel that route if I except more transfers? Maybe my approach is too rigid? 

Both websites only need a pass cover number if you want to book. Availability check can be done by anybody :)

If you give a travel date and departure time, I can check Brussel-Zuid - Marseille.

Accepting more transfers or being flexible about your travel date helps. Availability from Lille is usually better. You could take the IC to Antwerp, change to Lille and take a TGV to Marseille there.

The travel dates are 1 May for Amsterdam - Marseille and 8 May in the other direction. That already limits the possibilities because many people will be travelling (Sunday, holidays). Also, there are engineering works between Mouscron and Lille, which makes that route undesirable.

The TGV Brussels (13:17) - Marseille is already fully booked in 2nd class.
There are no passholder seats anymore in Thalys Amsterdam - Paris before 17:15. There is some availability Antwerp - Paris or Brussels - Paris.

Possible would be on 1 May:
Amsterdam C 10:28 - Antwerpen-C 12:16
Antwerpen-C 12:30 - Paris Nord 14:35 (25€)
RER D (2 stops) southbound; tickets at the Thalys bar (also buy them for the return journey).
Paris Lyon 15:37 - Marseille St C 18:57 (20€) (or 10€ one hour later)

On 8 May:
The direct TGV Marseille 12:10 - Brussel-Zuid has no passholder seats anymore.
Thalys Paris - Amsterdam does not have passholder seats anymore at useful times. Same for Paris - Antwerp. There is still some availability on Paris - Brussels.

Possible would be on 8 May:
Marseille St C 9:02 - Paris Lyon 12:21 (20€)
RER D (2 stops) northbound
Paris Nord 14:25 - Brussel-Zuid 15:47 (25€)
Brussel-Zuid 16:44 - Amsterdam C 19:35

Book the reservations in the next few days if you want to do the trip.

@rvdborgt 
Great!
Thanks a lot for the extensive research on this trips. It helped me a lot!

I still have a hard time finding the right information myself. I think I know how to check the availability of passholder seats on the TGVs on this website (is this website part of the Interrail website or of the NMBS website?), but the route from the France territory to Belgium (or Bruxelles), or The Netherlands (Amsterdam) is still a mystery to me. Seems it takes a lot of experience to easily find your way on the European railway network.

Anyway … many thanks again!