My turn signal is not blinking. How do I find the cause and repair it ? Also, How do I remove the turn signal lamp bulb ?

Make sure that all the bulbs are the correct ones and have a good ground.

The turn signals are controlled by the BCM
Turn Signal Lamps
The body control module (BCM) is responsible for controlling the turn signals and hazard lamps. When the turn signal switch is placed in either the left turn or right turn position, the BCM receives a signal from the turn signal switch. The BCM then sends an on-off voltage signal to either the left or right turn signals and their instrument panel cluster (IPC) indicator. When the hazard switch is pressed, the BCM receives a signal from the hazard switch. The BCM then will send an on-off voltage signal to all the turn signal lamps. All the turn signal lamps will flash including both IPC turn signal indicators.

I would suspect a problem with the BCM or maybe corrosion in the under hood fuse box . Wires from the BCM to the front turn signals go through the underhood fuse box.
Your best bet ,take it to a qualified repair shop that has a diagnostic scan tool that can read BCM DTC'S - diagnostic trouble codes
DTC B3948 Left Front Turn Signal Circuit
The body control module (BCM) controls the left front turn signal, hazard lamp, and daytime running lamp (DRL). The BCM supplies voltage to control the lighting functions. The BCM monitors the left turn signal lamp supply voltage circuit for proper operation.
DTC B3949 Right Front Turn Signal Circuit

Here is what the service manual says. Looks like it is not a flasher relay but a function of the body control module.

TURN SIGNAL AND HAZARD WARNING LAMP SYSTEMSystem Description OUTLINE Turn signal lamp and the hazard warning lamp is controlled by combination switch reading function and the flasher control function of BCM.
TURN SIGNAL LAMP OPERATION - BCM detects the combination switch condition by the combination switch reading function. - BCM supplies voltage to the right (left) turn signal lamp circuit when the ignition switch is turned ON and the turn signal switch is in the right (left) position. BCM blinks the turn signal lamp.
HAZARD WARNING LAMP OPERATION BCM supplies voltage to both turn signal lamp circuit when the hazard switch is turned ON. BCM blinks the hazard warning lamp.
TURN SIGNAL INDICATOR LAMP AND TURN SIGNAL SOUND OPERATION - BCM transmits the turn indicator signal to the combination meter with CAN communication while the turn signal lamp and the hazard warning lamp are operating. - Combination meter outputs the turn signal sound with the integrated buzzer while blinking the turn signal indicator lamp according to the turn indicator signal.
HIGH FLASHER OPERATION (FAIL-SAFE) - BCM detects the turn signal lamp circuit status by the terminal current value. - BCM increases the turn signal lamp blinking speed if the bulb or harness open is detected with the turn signal lamp operating. NOTE: The blinking speed is normal while operating the hazard warning lamp.

3 wing nuts. here is the info needed.

Service Information

2005 Buick LaCrosse ' Allure, LaCrosse VIN W Service Manual ' Body ' Lighting Systems ' Repair Instructions ' Document ID: 1537542 Tail Lamp, Turn Signal Lamp, High Mount Stop Lamp, License Lamp, Backup Lamp Replacement - Rear


Callout
Component Name
Notice: Refer to Fastener Notice in the Preface section.
Fastener Tightening Specifications: Refer to Fastener Tightening Specifications .
Turn Signal/Stop Lamp Assembly Replacement - Rear 1
Nut, Turn Signal/Tail Lamp (Qty: 3)
Tip
Pull back rear compartment carpet to gain access to lamp.
Tighten
2 N·m (18 lb in)
2
Lamp, Turn Signal/Tail Lamp
Tip
Twist the bulb socket and pull inward to replace the bulb.
License Lamp Replacement 3
Screws, License Lamp (Qty: 2)
Tighten
2 N·m (18 lb in)
4
Lamp, License Lamp
Tip
Twist the bulb socket and pull inward to replace the bulb.

Please check the cables lead to your rear turn lamp for loose connection. Both turning lamps ( rear and front) should work to keep turning lights back to normal blink.
Regards

Many turn signal lamps will have multiple bulbs in them. If "one" of the bulbs is faulty, the interior lamp or lamps will come on, indicating a bad bulb. You will need to turn on your left blinker and view the bulb and make sure all bulbs appear to be blinking on front and rear of the car, then repeat with the right blinker. If you find one that doesn't appear to look right or blinks faster slower or dimmer, then replace the bulbs in that turn signal. The interior light may begin working properly when you replace the "shorted" bulb for the turn signals. If it does not, check the fuses to make sure the "shorted" bulb didn't damage your fuse for the interior lights.

Tail Lamp: Service and Repair
TAIL LAMP REPLACEMENT
REMOVAL PROCEDURE
1. Open the rear compartment lid.
2. Remove the rear compartment trim panel (4).
3. Disconnect the tail lamp electrical connectors.
4. Remove the following components from the vehicle:
- The 3 nuts (3)
- The tail lamp (7)
5. Remove the following from the tail lamp:
- The stop lamp socket (2)
- The turn signal lamp socket (5)
6. Remove the following from the lamp sockets, if necessary:
- The stop lamp bulb (1)
- The turn signal lamp bulb (6)
INSTALLATION PROCEDURE
1. Install the following to the lamp sockets, if removed:
- The stop lamp bulb (1)
- The turn signal lamp bulb (6)
2. Install the following to the tail lamp:
- The stop lamp socket (2)
- The turn signal lamp socket (5)
3. Install the tail lamp (7) to the vehicle. Secure the tail lamp (7) with the 3 nuts (3).
NOTE: Refer to Fastener Notice in Service Precautions.
Tighten the tail lamp nuts to 10 N.m (89 lb in).
4. Connect the tail lamp electrical connectors.
5. Install the rear compartment trim panel (4) to the vehicle.
6. Close the rear compartment lid.

The bulb sockets are melted (not the bulb) - go to the junk yard and pick up a couple and keep a couple in your glove box - Have the same problem.

Gregg -

I had the same problem. It was a defective lamp socket on the driver's left side, Volvo Part Number 8662985. Costs me $11.24 to replace the lamp socket, which included the bulb. If you've already replaced the bulbs yourself, you will find replacing the socket pretty easy, just use a thin flathead screwdriver to delicately pry off the metal retainer clip which holds the lamp socket in place, as you'll need to reinstall the existing metal retainer clip once you've replaced the lamp socket. Two minute job.