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I am trying to make a smooth, well meshed corner from a sphere with 3 extensions at right angles. As you can see from the image that the mesh is 'messy'. Any suggestions on how to do this better?

I start with a uv sphere, proportional grab from a pole, then duplicate and rotate the 3 extrusions into position and union the result. I've tried this multiple times but always end up with a mesh that is less than perfect.

I've realized that a uv sphere has 2 poles so clean grab of 3 'sides' is tricky.

I've followed a number of tutorials but am now venturing off into modelling my ideas and I'm finding all kinds of interesting challenges.

I've tried similar with an ico sphere and a subdivided (rounded) cube but I've not been happy with any of the results. I've also tried uv spheres of different segments and rings.

The left hand side shows the mesh inside the sphere, the right hand side show the render view outside with red circles around the problem areas.

Any advice or ideas are welcome.enter image description here

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It's not clear why you're not satisfied with a subdivided cube, you can subdivide it 3 times with a Smoothness of 1:

enter image description here

Cut off 3 holes, make the holes circular with AltShiftS:

enter image description here

Extrude:

enter image description here

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