Here's a list of what can and can't be planted together. I haven't tried most of these but this is what I found last year from a bit of online research. I've also got a list og herbs which I add at some point. It would be useful to know if anyone agrees with this or has tried these combinations.
Asparagus
Companion Plants: Tomato, Parsley, Basil
Incompatible with:
Beans
Companion Plants: Most Vegetables & Herbs
Incompatible with:
Beans, Broad
Companion Plants: Potato, Cucumber, Maize, Strawberry, Celery, Summer Savory
Incompatible with: Onion
Beans, Runner
Companion Plants: Maize, Summer Savory, Radish
Incompatible with: Onion, Beets, Kohlrabi, Sunflower
Cabbage Family
Companion Plants: Aromatic Herbs, Celery, Beetroot, Onion Family, Chamomile, Spinach, Chard
Incompatible with: Dill, Strawberries, Runner Beans, Tomato
Carrots
Companion Plants: Pea, Lettuce, Rosemary, Onion Family, Sage, Tomato
Incompatible with:
Celery
Companion Plants: Onion & Cabbage Families, Tomato, Broad Beans, Nasturtium
Incompatible with: Dill
Cucumber
Companion Plants: Beans, Maize, Pea, Sunflowers, Radish
Incompatible with: Potato, Aromatic Herbs
Aubergine (Eggplant)
Companion Plants: Broad Beans, Marigold
Incompatible with:
Lettuce
Companion Plants: Carrot, Radish, Strawberry, Cucumber
Incompatible with:
Maize (Corn)
Companion Plants: Potato, Broad Beans, Pea, Pumpkin, Cucumber, Squash
Incompatible with: Tomato
Melon
Companion Plants: Maize, Nasturtium, Radish,
Incompatible with:
Onion Family
Companion Plants: Beetroot, Carrot, Lettuce, Cabbage Family
Incompatible with: Beans, Peas
Parsley
Companion Plants: Tomato, Asparagus
Incompatible with:
Peas
Companion Plants: Carrots, Radish, Turnip, Cucumber, Maize, Beans
Incompatible with: Onion Family, Gladiolus, Potato
Potato
Companion Plants: Beans, Maize, Cabbage Family, Marigolds, Horseradish
Incompatible with: Pumpkin, Squash, Tomato, Cucumber, Sunflower
Pumpkins
Companion Plants: Maize, Marigold
Incompatible with: Potato
Radish
Companion Plants: Pea, Nasturtium, Lettuce, Cucumber
Incompatible with: Hyssop
Spinach
Companion Plants: Strawberry, Cauliflower, Celery
Incompatible with:
Squash
Companion Plants: Nasturtium, Maize, Marigold
Incompatible with: Potato
Strawberry
Companion Plants: Broad Beans, Lettuce, Onion, Spinach
Incompatible with: Cabbage
Tomato
Companion Plants: Basil, Onion Family, Nasturtium, Marigold, Asparagus, Carrot, Parsley, Cucumber, Mint
Incompatible with: Potato, Fennel, Cabbage Family
Turnip
Companion Plants: Pea
Incompatible with: Potato