I'd second Clive's comments - classic symptoms of poor contacts. I know you cleaned them, but did you clean the important bits?
General comments about cleaning (ie not saying anyone in particular hasn't done it properly!!). It's an unfortunate fact that our eyebrow tends to sit very neatly in the camera hotshoe when it's not otherwise occupied and this transfers all sorts of !Yuk! into the hotshoe. That and residual manufacturing muck can be sufficient to prevent a good contact - remember we are talking low voltage circuits here!
Here are the bits that 'cleaning' often misses.
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Another point Clive raised concerns proper seating of flash or trigger. Many folks think of the retractable pin on a hotfoot as being a locking pin. It's not - or rather, that is a secondary function. These pins across all makes
have a very light spring loading. Merely sliding the hotfoot into the hotshoe and tightening down is absolutely NO guarantee that the pin has sprung into the hole. Think of it rather as a LOCATING PIN. If you push the flash/trigger into the hotshoe and HALF wind the pin down - then try to pull the device out of the shoe. If it won't move, that indicates that the pin has accurately located and dropped into it's corresponding hole. Then you can screw down tight secure in the knowledge that the foot really is fully located and your contact pins are doing their job properly.
If neither of these solutions fix the problem for you, then it looks like you have a fault with the camera hotshoe. The last thing you can do is take a close look at the camera hotshoe. From camera back, under the left rail, is a tiny microswitch which tells the camera that there is something in the shoe. It has very occasionally been found that this switch is a tiny bit loose and not triggering properly. This would lead to intermittent work/no work of the external flash menu (and the flash or trigger).
If the first two solutions don't solve the problem, I reckon it's the dreaded "RTB" to get the camera fixed. It'd be handy, as a final check, to try your flashes and triggers on a different camera if you can. Perhaps in the shop where you bought it if that's even remotely possible?
Edit: Clarification re locating pins across all makes.