Grid Selections in QGIS

There are several possibilities to achieve this - this is a rather straight forward one, with probably a lot of manual work.

Run join attributes by location: Your first layer is the grid, your second layer your Counties. Either attach all attributes of the Counties to the grid, or choose only their name. Make sure to set one to many as your join-type.

Then use select by attribute on the modified grid and go through each County name - save the selection to a new file, then select the next name.

This probably could be done faster using Python or maybe SQL.

This can be done quite simply and easily using one of QGIS's built-in tools, Extract By Location. Not only does this tool subset your input layer by its spatial relationship to another layer, but it can do so on a per-county basis.

If I've understood the process in your question, proceed as follows.

1. Open the Extract By Location tool.
2. Specify your_grid_layer under Extract features from.
3. For the geometric predicate, Intersect should be selected by default, and should work well enough, but you can change this as needed.
4. Specify your_counties_layer under By comparing the features from.
5. Most importantly! Click the green circular arrows to iterate over the counties layer. This will create a separate extracted output for every county feature.
6. Run the tool.

Input layers and tool:

Output layers:

Note that each color represented in the image above is a separate layer, I've simply chosen to match their colors to the input features. And in each, the layer corresponds to the cells of the input grid which intersected a single county feature.

Also! Because many grid cells intersected more than one polygon, there is considerable overlap among these output layers. With only two of them on, this is more apparent.

If you just want to make a selection, you could also use Select by location.

1. Go to menu Vector > Research tools > Select by Location.
2. Select your grid layer as input and check one of the geometric predicates (I used intersect). By comparing to the features from: chose your boundaries shapefile layer (where you made your original selection). Check the Use selected features only checkbox and be sure to define Modify current selection by creating a new selection.

That's all. See screenshots:

So it turns out to be quite straightforward. I don't profess it's a panacea but it seems to work for the project in question. Sincere apologies for the non-technical layout below but they're hopefully a useful set of instructions.

Please post any questions. Huge thanks again to the original respondents.

Instructions

The following operations are all accessed through the Processing Toolbox panel:

Step 1: Isolate county boundary to work on: County Boundaries - Attribute Table - Select - Invert Selection - Delete

Step 2: Build vector grid: Vector Creation - Set Grid - Polygon – Enter bespoke Extents and Spacings

Step 3: Extract Grid Squares that intersect with county boundary: Vector Selection - Extract by Location - Extract from GRID (from Step 2) by COUNTY (from Step 1)

Step 4: Clip Grid Squares to County Boundary: Vector Overlay - Clip - Input: EXTRACTED BY LOCATION (from Step 3), Overlay: COUNTY (from Step 1)

Step 5: Prepare Clipped layer for labelling individual cells – Stage 1: Vector Table - Add ‘autoincremental’ field - Input: CLIPPED (from Step 4), Field Name=AUTO, Start Values=1

Step 6: Prepare Clipped layer for labelling individual cells – Stage 2: Vector Geometry - Centroids - Input: INCREMENTED (from Step 5)

Step 7: Sequential labelling of individual cells: Layer Properties of CENTROIDS (from Step 6) - Single Label – AUTO (see Step 5) - Placement: Around Point

Step 8: Select features from within individual cells: Vector Overlay – Intersection – Input: features layer (from your project), Overlay: CLIPPED LAYER (from Step 7) – Open Attribute Table for newly created layer - Sort Results by AUTO column (or any mode of analysis you choose).

E & O E