My '95 Olds Achieva did not pass safety this year because the high beams won't turn on. I have checked the fuses and they are all fine. I have no idea where to begin with finding a solution to this.

Lets start with the basic info I need to help, the make, model and year.

top middle connection is the earth on the bulb and this is faulty ,what end i couldnt tell you but its no doubt the chassis side so run another cvable to teh neg on the battery and if it brightens up then you know its a earth problem

Check the ignition switch connector. Check for 12v on the start circuit with the ignition in the start position. Check neutral safety switch connector. Check for 12v in and out of the N/S switch when the gear selector is in neutral or park and the ignition is in the start position. If you have 12v there check at the small terminal on the starter solenoid. If you find 12v there and on the big terminal there would appear to be a starter problem.

Common problem, there is a headlight relay in the power distribution center under the hood that needs changed out.

How to Replace a Headlight Bulb
(DIY Complexity: Easy / Time: 0.9 hour)

Parts:
1. Headlight Bulb

Tools:
1. Screwdriver
2. 1/4 in. Drive Ratchet
3. Socket Set



HIGH PERFORMANCE HALOGEN HEADLIGHT BULB -- Low 45/high 65 watts, Dual C-8 filament for both high beam and low beam; 100 percent legal halogen headlight bulbs; Features the whitest and brightest light color similar to daylight for safer night driving; Meets all US FMVSS 108 and Canadian CMVSS 108 requirements.

Steps:
Step 1 Open the hood and locate the headlight assembly.
o Tip: Most headlight assemblies have a place in the rear to gain access to the headlight bulb.
o Tip: It is recommended that headlight bulbs are replaced in pairs.
o Tip: Always make sure the struts on the hood can fully support the weight of the hood so that it does not close on you while working. The use of a hood prop can securely hold it in place.
o Tip: Safety Tip:Always wear safety glasses when working on your vehicle. Wear other personal protective equipment (PPE) when necessary, for example latex gloves or safety shoes.

Step 2 Gain access to headlight bulb.
o Tip: It is a good idea to check the high beam and low beam headlight bulbs at this time.

Step 3 Disconnect the wiring harness and replace the old headlight bulb with a new one.



Remove the retaining ring from the base of the bulb, then...




...turn the bulb and remove it from the headlight assembly.

Step 4 Turn on the headlights and check high and low beams to verify repairs.


Hope this help (remember comment and rated this).

Your Olds has 2 flasher relays. One for the turn signals and one for the hazard lights.
The turn signal flasher relay is located on the driver's side, under the dash, mounted on the steering column.
The hazard flasher relay is located driver's side, under dash, driver's side of steering column, mounted on fuse panel.

Remove and replace flasher.

You know the fuse for the high beams is good ( Cab fuse panel fuse # 16 ) and the pass low beam fuse ( cab fuse panel # 26 ).
Might want to check cab fuse panel fuse #28 ( this is the driver's side low beam ).

If this fuse has power on both sides ( with the low beams on ) next you can pull the headlamps, and check for power at the connectors, but depending when the last bulb change took place, I would say you need need bulbs. ( need at least 1 for the high beam out )
Confirming with a meter at the headlamp would be a good idea, before spending the money.

The bulb connector has 3 pins.
Black wire is ground on both sides
Light Green w/ Black stripe wire is high beams on both sides
Driver Lo = Dark Blue w/ white stripe
Pass Lo = Dark Green w/ Orange stripe.

The click you hear is the starter relay. Check for continuity thru relay with battery negative cable disconnected and switch in start position. If no continuity, it probably has burned contacts. Sometimes these can be taken apart and contacts filed/sanded cleaned.

probably the bulbs.