Similar to the answer provided by @csk - I would:
1. Use the Merge vector layers tool to combine all the catchments into a single layer.
2. Use the Join attributes by location (summary) tool to count the OBJECTID (but could be any unique ID column) of the past landslides that intersects the catchments.
This will create a new catchments layer that contains ...
Use a value other than 0.0 for the burn-in value, such as 1.0.
Use the extent of the land cover raster or the 10km grid
Change output raster size units to map instead of pixel,
Specify 10000 (your map units are metres, i.e 10km) for the resolution.
How about something like:
array_contains(string_to_array("Class", ','), '2')
This splits the string in the field "Class" into an array and returns true if the array contains the given value ('2'). This will give you all the features that contain the class/integer 2 (and leave out 12).
Tested on QGIS 2.18 and QGIS 3.4
I can suggest using a "Virtual Layer" through Layer > Add Layer > Add/Edit Virtual Layer...
Let's assume we have ten features in "multiple_polygons" (yellow), and three in "catchment_zones" (grey) accordingly, see image below.
With the following Query, it is possible to achieve the result
SELECT c.geometry, c.id, c....
Also you can use aggregate expression in Field Calculator and create new field with interesection count, similarly to my answer on QGIS how to sum a certain vector field inside a polygon and populate a field in the polygon?. Instead of sum aggregate function use count like this:
expression:= $id ,
After your comment, I double checked, and this is what I worked out. I start with a multipolygon multi with three areas and two attrib columns cat (value 1) and label (value "one").
# The initial multi_polygon
GRASS 7.6.0 (WGS84):/tmp > v.info -t multi
Based on this forum response here How to intersect a raster and vector
You can use the Raster to Polygon tool. Which can result in many shapefile/polygons but in your case may not be an issue
Converts a raster dataset to polygon feature
An alternative is using the Zonal Statistics
Calculates statistics on values of a raster within the zones of ...